The planets of the solar system are named after gods and goddesses and are known by their Roman, not Greek names. Here I'll put the Greek equivalents in brackets afterwards. The satellites of these planets for the most part tend to be the Greek names associated in one form or another with the parent body or the god/goddess in question. Moons (and major asteroids) are identified with an asterisk.
Now you need to understand that at the time of the ancient Roman or Greek pantheon, only eight of the solar system's celestial bodies were known. There were the three major bodies, the Sun, Moon, and Planet Earth itself of course (thus, way back then, you'd expect a mythological relationship between these three bodies), and the minor five visible planets - Mercury, Venus, Mars, Jupiter and Saturn which were just points of light to the ancients, though it's not surprising the ancients matched celestial physics with the personalities of their gods and named them accordingly. Apart from our satellite, the Moon, no other moons were known to the ancients.
MYTHOLOGY AND THE INNER SOLAR SYSTEM
SOL (Helios) is obviously the god of the Sun. It surprises me that given the prominence and importance of this celestial object, our Sun, that the god of the Sun didn't take pride of place as head of the gods. Be that as it may, Sol, a first generation Titan, was born of the god Hyperion and the goddess Theia, brother to Luna (Selene) and Aurora or Dawn (Eos). His job is to bring light to men and gods alike. He rides in the sky in a fiery chariot pulled by his team of fast horses. Sol, from his vantage point, sees and hears everything (which proved to be the undoing of an illicit relationship between Mars (Ares) and Venus (Aphrodite). The god Apollo later fused as one with Sol's image though once they were two different deities. Sol's son, Phaethon, by the ocean nymph Clymene, once tried to ride his father's horse-drawn chariot, but lost control and had to be killed by Jupiter (Zeus) before the runaway Sun destroyed everything.
VULCAN (Hephaestus) was once upon a time actually sighted by astronomers and as the closest planet to the Sun, it would of course be HOT, and thusly was named after the deity of fire and blacksmithing. Alas, Vulcan turned out to be as mythical as the deity it was named after.
MERCURY (Hermes) when observed from Earth goes around the Sun quick-smart. So, it was natural to name that body after the fleetest of the gods, the messenger god who flies through the skies on a winged helmet and winged sandals, Mercury. No great mystery there.
VENUS (Aphrodite) was, because of its celestial purity of face and brightness, a female goddess, and of course THE female goddess was the goddess of love. Alas, Venus the planet turned out to be more appropriate for the Christian ruler of the underworld than the goddess of romance. Venus is Hell personified and thus the reality turned out to be vastly different than the expectation, but the ancients couldn't have known that back then. Venus (or Aphrodite) had an unusual birth, the offspring of Gaia, but fertilized by the blood/semen of Gaia's son and lover, Uranus, when Uranus was castrated by Saturn (Cronus).
*Neith was the name given to the lone phantom satellite of Venus. I say phantom because, like Vulcan, it was a case of 'now you see it, now you don't'. It vanished from the sight of astronomers across the world, but prior to that some saw something. That's water under the bridge now. The naming of the satellite Neith broke from tradition for Neith was the mysterious early Egyptian goddess of Sais. She was identified with war and hunting and the personification of the primordial waters of creation, though later on down the track she was associated with weaving, which is what Neith means. Wisdom was one of her strong points. She had no known male companion, so was known as a 'Virgin Mother Goddess'.
GAEA (Gaia) otherwise known as Terra or Earth was one of the original deities 'born' out of the original state of the universe - Chaos. Given the relative lack of bed partners around then, she mated with any and all available males, including her asexually created children, like Uranus.
*Luna (Selene) was born of the god Hyperion and the goddess Theia, sister to Sol (Helios) and Aurora or Dawn (Eos). Luna crosses the night sky on a chariot pulled by two white horses. As with Sol and Apollo, the Roman goddess Diana (Artemis) is often associated with, and has melded with Luna (as the crescent-shaped Moon) as has the goddess of the underworld, Hecate (as the New Moon). Luna remains as the Full Moon in the night sky.
MARS (Ares) son of Jupiter (Zeus) and Juno (Hera or Mrs. Zeus) was the god of war, though if mythology be believed, he was rather a wimp of a god, got his butt kicked on several occasions and was rather disliked up on Olympus.
*Deimos: Ares (the god of war) had an extramarital fling with the goddess of love, Aphrodite (who was married to the god of fire, Hephaestus) and from this union produced two sons, one being Deimos, meaning 'fear'.
*Phobos: That Ares and Aphrodite pairing also produced another son, Phobos, meaning 'terror'. Both Deimos and Phobos often joined daddy on the battlefield. However, apart from that, they play no significant role in mythology.
ASTEROIDS: There are only four really main or large asteroids among the tens of thousands that exist, Ceres, Pallas, Vesta and Juno.
*Ceres (Demeter) was the goddess of agriculture, the harvest, and the one who controlled the seasons. Zeus had a bedroom fling with her that resulted in Persephone, who Zeus then promised to his brother Hades as his wife. Alas, Zeus didn't inform either Persephone or Demeter about that decision, which had major repercussions down the track.
*Juno is Mrs. Jupiter, or in the Greek, the equivalent of Hera (Mrs. Zeus). In either case, Juno/Hera is the queen of all the gods and goddesses. She is perhaps noted most of all for her vindictiveness against some of the offspring produced by her husband's extramarital relationships, most notably, Hercules.
*Pallas is named after Pallas Athena, an alternative name for the Greek goddess Athena. In another version Pallas was the daughter of Triton, who acted as foster parent to Athena. A friendly fight between the two goddesses went sour and Pallas was killed.
*Vesta (Hestia) was the virgin goddess of the hearth and home or home fire, a real stay-at-home sort of deity and thus doesn't feature much in mythological adventures and doings. In the Roman version Vesta was represented by a shrine with an eternal flame guarded by the Vestal Virgins who, as a condition of their employment and of remaining alive, had better remain pure, or else. The god of war, Mars, raped one, and from that involuntary union, sprang the twins Remus and Romulus, the latter the founder of Rome.
MYTHOLOGY AND THE OUTER SOLAR SYSTEM
JUPITER (Zeus) was king of the mountain - Mount Olympus that is. He, after a series of 'wars in heaven' (with the Titans) and on earth (with the Giants) and finally with the daddy of all monsters, Typhon, claimed the right to be worshiped as King of the Gods, by all lesser gods and goddesses, demigods and demigoddesses, as well as all of the great unwashed - the mortals comprising humanity. Apart from being fast on the draw with his trademark thunderbolts, he's most noted for being even faster on the draw with his private parts. No Mounties or bounty hunters ever perused their quarry better or with more determination than our always horny Jupiter/Zeus, as we shall see immediately below (though with over five dozen satellites, only a few can be entered into in this short essay).
*Amalthea: Jupiter (Zeus) as a young infant, was spirited and hidden away in Crete by his mother, Rhea from his daddy, Saturn (Cronus) because daddy didn't want any sons of his eventually growing up and challenging him for supreme power. Saturn thus kept them out of his harms way by swallowing them. Rhea saved the last (Jupiter) by trickery. Anyway, young Jupiter was reared and looked after by local nymphs on Crete, especially by Amalthea. She's also known for having a horn, the 'Horn of Plenty' a cornucopia that provided limitless food and drink for whoever possessed it.
*Ananke, the goddess of inevitability and personification of destiny, necessity and fate was the mother of the Moirai by Zeus. The Moirai, also known as The Fates, were the trio that spun the thread of life of mortals from birth to death and thus controlled everyone's destiny. Even Zeus was powerless against their will.
*Callisto was the daughter of the King of Arcady, and she had the misfortune to have Zeus fall for her. Callisto was associated with the virgin goddess Artemis. When Zeus raped Callisto, she got in a family way (in mythology, every rape results in a pregnancy otherwise there's no point to the rape in the first place). This rape ultimately produces a son, Arcas. Artemis, being the virgin, was livid and banished Callisto from any further association with her. Hera (Mrs. Zeus) in the meantime assumed Callisto was a willing partner to her wayward hubby and in revenge turned Callisto into a bear. Years later, Arcas, out hunting, ran across that bear and not realizing that it was mom, shot her dead. Zeus then placed Callisto into the zodiac as Ursa Major, the Great Bear (or the Big Dipper); Arcas ultimately got his place in the heavens too as the Little Bear (or the Little Dipper).
*Carme was the mother by Zeus (who else?) of the virginal huntress Britomartis, a Minoan or Cretan goddess. Carme assisted in the harvest of the grain.
*Elara was the mother of the giant, Tityos. Tityos was fathered by you know who - randy Zeus. Tityos met a bad end, killed on the behest of Hera by Artemis and Apollo, and forever tortured in Tartarus (a subdivision of Hades) by a couple of vultures who liked liver for breakfast, lunch and dinner. Fortunately for the vultures, Tityos's liver kept regenerating so they never lacked for food.
*Europa was the daughter of a Phoenician king. Zeus got the hots for her and turned himself into a snow-white bull (gods can shape-shift; it's one of their major superpowers and that's no bull) and approached Europa in that form. Europa playfully climbs on the Zeus-as-bull's back. Zeus-the-bull then dives into the sea with Europa on his back and swims out to Crete. Once there he reverts back into his 'human' form and they made mad passionate love together and in the fullness of time Europa gave birth to triplets, one of which she named Minos. Zeus, ever thoughtful, also gave her some other gifts as well for her troubles. Europa ultimately married the king of Crete. Ultimately her son, Minos, assumed or claimed the throne. That's why Crete's ancient civilization is called the Minoan society. Europa is the only person having an entire continent named after them - Europe. As an aside, Europa is one of the few celestial bodies of interest to those on the lookout for potential sites where extraterrestrial life might exist.
*Ganymede: Abductions by the gods are by no means unknown or uncommon. In Greek mythology, a young handsome lad was Ganymede. He was abducted by Zeus, shape-shifted in the form of an eagle. Zeus favored the ladies, goddesses, demigoddesses, even mortal women, but for a catamite change of pace now and again... Ganymede was rewarded for his services by eternal youth and immortality, eventually placed in the sky as the constellation Aquarius. His daddy (Tros) was also compensated with a gift of fine horses.
*Himalia was a nymph who bore a trio of sons to Zeus or of course the Roman equivalent, Jupiter.
*Io was the daughter of the first king of Argos (what's it about kings and their daughters already - that plot device has reached its use by date). She of course caught the lustful eye of Zeus. Hera (Mrs. Zeus) got suspicious, so Zeus turned Io into a cow as a disguise. Hera (knowing full well what was afoot) of course assumed the cow was a gift for her from her hubby since he obviously had no use for a cow, so Zeus had no option but to agree that that of course was what he had intended all along. Hera then had the cow guarded by the multi-eyed giant Argus. Zeus, needing to get Io back, got Hermes (Mercury to the Romans) to kill Argus. Hera in revenge plagued Io-the-cow with a hornet. Finally a truce was called. Zeus promised to behave (pull the other one) and Io was reverted back to human form and reunited with her worried daddy - the king.
*Leda was of course the daughter of a king, and in turn married a king, in this case a king of Sparta. Zeus, of course, took a shine to Leda, and took the form of a swan and flew into Leda's arms seeking 'protection' from another predatory bird - that's his story and he's sticking to it. Anyway, somehow or other (Zeus may have turned her into a goose) this cuddle made Leda pregnant and from this pregnancy she gave birth - she laid an egg, obviously! The egg hatched and out popped beautiful Helen. Helen, later known as Helen of Troy, was the face that launched a thousand ships (and then some). This was Zeus's contribution to the Trojan War. In another version, the egg was the union between Zeus and the goddess of revenge, Nemesis. They gave the egg to Leda to hatch and raise the offspring up as her own. Leda had another child by Zeus - one of a pair of twins actually, Castor (by Zeus) and Pollux (by her husband) but known as the 'Boys of Zeus' or the 'Heavenly Twins' who eventually sailed with Jason as part of the Argonaut crew.
*Lysithea was the daughter of Oceanus (a Titan, product of Gaia and Uranus) and one of Zeus's many, many lovers.
*Pasiphae was the wife of King Minos of Crete, who, via a rather strange mating arrangement gave birth to the Minotaur. The rather remote connection to Jupiter is that Jupiter's brother, Neptune, was behind all of this. Neptune was offended because Minos didn't offer up as a sacrifice to Neptune a favorite bull, so in retaliation, Neptune saw to it that Minos's wife became infatuated with, and lusted after a bull, the offspring being that well known hybrid, the Minotaur.
*Thyone was once known as Semele, a lover of Zeus, who produced as offspring Dionysus, the god of wine and all around fun times. When Semele died as a result of Zeus revealing to her his true appearance - all the result of Hera's trickery - and went to Hades, Dionysus rescued her from the underworld and brought her to Olympus, made her into a goddess along with providing the change of name.
SATURN (Cronus) was one of those first generation Titans, offspring of Gaia and her son Uranus. He mated with Rhea, and produced from that union the original half-dozen gods and goddesses that would eventually become the Olympians. But the transition from Saturn's offspring to become the rulers of Mount Olympus was an epic that could just about out-epic anything Hollywood has ever done. Saturn is known for two things in particular - castrating his daddy and snacking on his kids - Yum, yum.
*Dione is a very obscure goddess who may have been a Titan or a Nereid or an Oceanid. Her name seems to be a feminine form of Zeus, and that brings in the alternative version of the origin of Aphrodite - a traditional birth via the union of randy Zeus and Dione (whoever she was). At least she's now immortalized as a satellite of Saturn.
*Enceladus was one of the Gigantes. The Gigantes were offspring of Gaia, fertilized by the blood/semen of her son and lover Uranus who was ultimately castrated by Cronus (a son of Gaia and Uranus back when he had his private parts intact). In the Gigantomachia (the battle between the Olympian gods and the giants), Athena deposited Enceladus under Mount Etna and the fire and brimstone breath of Enceladus accounts for that volcano's well, fire and brimstone. This is another abode thought as a possible site where alien life forms might one day be discovered.
*Hyperion was a Titan god, born of Gaia (Gaea) and her son, Uranus. He would become an early god of the Sun.
*Iapetus was another Titan god, born of Gaia (Gaea) and her son, Uranus. He would become the father of Prometheus, who ultimately is remembered for stealing fire from the gods and giving it to mankind.
*Janus was the Roman god of beginnings, doorways, entrances, gateways, etc. and is depicted with two faces, one each facing forward (to the future) and the other backward (to the past). January, the beginning, is named after him.
*Mimas, like Enceladus, was one of the Gigantes of Greek mythologies. Like the other giant sons of Gaia and Uranus, Mimas had serpents for legs and was born fully armored. Mimas was slain by Hephaestus during the war against the Olympians by a volley of molten iron.
*Phoebe was a Titan goddess, born of Gaia (Gaea) and her son, Uranus. Phoebe is identified as an early moon goddess. In another context, the goddess Artemis sometimes was associated with an alias of Phoebe. Finally, Helen of Troy had a half-sister Phoebe.
*Rhea was another Titan goddess, born of Gaia (Gaea) and her son, Uranus. She married Cronus (Saturn) and from that union was born what ultimately became the original half-dozen Olympian gods and goddesses. Cronus, out to eliminate potential future rivals, swallowed the first five of these kids, but Rhea tricked him when it came to the sixth and last, Zeus. She spirited Zeus away to Crete to be raised, and tricked Cronus into swallowing a stone disguised as the baby Zeus.
*Tethys was yet another Titan goddess, born of Gaia (Gaea) and her son, Uranus who would become the most ancient goddess of the sea.
*Titan is the largest moon of Saturn, a moon of interest to astrobiologists, and the only natural satellite to have an extensive atmosphere. In fact, Titan is larger than the planet Mercury and way lager than our own satellite, Luna. The mythology connection is that the original dozen Titans (some of which are named satellites of Saturn) were one of a set of offspring between Gaia and Uranus. Among the second generation of Titans are several household names like Atlas and Prometheus. They were overthrown in the Titanomachy (The War of the Titans) by Zeus and his fellow Olympians.
URANUS (alternative spelling Ouranos) was the original sky god, the son, via parthenogenesis, of Gaea (Gaia) who shacked up with mum to produce all manner of deities, including monstrous ones. He's known for having his privates sliced off by one of their Titan offspring, Saturn (Cronus). Saturn in turn got done like a dinner by one of his offspring, Jupiter (Zeus) but that's another story.
*Ariel, Miranda, Oberon, Titania and Umbriel, the five major moons of Uranus, were named or chosen from the names of characters from the works of Shakespeare and Alexander Pope and thus, unfortunately, have near bugger-all to do with traditional Greek and Roman mythology. Bummer, as that spoils some potentially good stories relating back to and involving the parent god, Uranus.
NEPTUNE (Poseidon) was the god of the sea, one of the original half-dozen Olympian gods and goddesses, brother to Zeus, Hades, Hera, Hestia and Demeter. All of Neptune's known 13 satellites are named for Green and Roman water deities or groups of minor water gods. The major two are Nereid and Triton.
*Nereid - the Nereids (plural) were nymphs of the sea, the 50 daughters of the Mediterranean Sea god Nereus. Among the famous of the Nereids - goddess all - were Thetis (mother of Achilles), Galatea and Amphitrite (who married Poseidon).
*Triton was another sea god, a son of Poseidon and the Nereid goddess Amphitrite, a nymph of the sea who became Queen of the Sea when she wed Poseidon. The typical depiction of Triton is that of a merman - human head and torso; one or two fishy (or maybe dolphin) tails. He too has a trident and a conch shell which when blown can calm the stormy waves. Like father, like son, Triton also had a bit of an eye for the ladies, especially beach-babes. Triton favors in a positive light for assisting Jason and the Argonauts in their saga.
PLUTO (Hades) was god of the underworld. In the Greek version, Hades (the god) administered Hades (the underworld), so the name of the ruler and the place ruled were one and the same. Within recent times, Pluto was officially downgraded from planet to dwarf planet status, which made a lot of traditionalists very unhappy.
*Charon was the son of Erebus (the god of underground darkness) and Nyx (goddess of night), who played the role of the ferryman who rowed the newly deceased across the River Styx to Hades, the underworld. As a reward, or a fare, he got a coin in payment - no coin, no crossing. The parentage and place of employment are ideal for a satellite so far out from the Sun.
*Hydra (discovered in 2005) was named after that famed beastie slain by Hercules as one of his dozen labors in Greco-Roman mythology. Hydra was one of numerous monstrous offspring from Typhon and Echidna, themselves monsters.
*Nix (a deliberate alternative spelling of Nyx) was a Greek goddess of night, one of the original deities conceived from the original state of the universe, Chaos. Nyx was the mother of Charon. Like Hydra, Nix the satellite was discovered in 2005.
ERIS is another dwarf planet, a trans-Neptunian object even larger that Pluto discovered in 2005. In fact it's the most massive of all dwarf 'planets' in our solar system, even though it's three times farther removed from the Sun than Pluto. Eris is a Greek goddess, the personification of strife and discord.