I only know what most people are probably aware of:
Minerva: Roman Goddess of Wisdom
McGonnagall: One of the most abysmal poets of the English language was a Scot named William McGonnagall
Argus: Watchman with a hundred eyes. Servant of Hera. Some sources say he had eyes all over his body. He was sung to sleep by Hermes
Remus: Romulus and Remus were twin brothers, thrown into the Tiber. They were rescued by a she-wolf, who reared them. Later the brothers built a settlement. Remus mocked Romulus over the height of the walls, so Romulus killed him and named the settlement Rome, after himself.
Lupin: The French for wolf is 'loup'. Lupin comes from the Latin, meaning wolf. Lupus is a constellation called the wolf.
Sirius: (all together now) the Dog Star
Regulus: interestingly enough, Regulus means Prince or Heart of the Lion, and is the name of a star
Bellatrix: another star. Means warrioress, and is part of Orion, the hunter
Narcissa: Female version of Narcissus, the youth who was so vain that he fell in love with his own reflection
Draco: Serpent. Also a constellation called the dragon.
Rabastan: star meaning head of the snake. Found in the constellation Draco.
Andromeda: star constellation. She was a princess who was chained to a rock as a sacrifice to a sea monster, and who was rescued by Perseus.
Dumbledore: Famously the old word for a Bumblebee
Hermione: a Queen in Shakespeare's 'A Winter's Tale', who turns into a statue. Hermione is also the name of Helen (of Trojan fame)'s child.