Eurydice is based upon the figure of the same name from Greek myth. She was reputed to be a granddaughter of Ares and betrothed to Orpheus, the bard of Greek myth, and the son of Oneiros and the muse Calliope.
According to the retelling of her tale in Sandman, on the day of her wedding, Eurydice was assaulted by the drunken satyr, Aristaeus. When she fled from him, she stumbled into a nest of snakes and died from snakebite. Overcome with grief, Orpheus petitioned Oneiros' help to revive Eurydice, but disavowed his father when Dream proved unwilling to help.
His uncle, Olethros and aunt, Teleute proved a bit more cooperative and sent Orpheus to plead his case before Hades, the ruler of the underworld. He succeeded in convincing Hades to return Eurydice under the condition that he does not look back during the trek to the surface.
Unable to help himself, however, Orpheus looked over his shoulder when only steps from the surface and Eurydice was snatched away and taken back into the Underworld forever.