I don't believe "Fantine" is a proper name, that is no mother ever names a child "Fantine" it's more of a nickname. It comes from the same root as "infant". It basically means "babyish"... which being an orphan girl matches her character.
This name was used by Victor Hugo for the mother of Cosette in his novel 'Les Misérables' (1862). The name was given to her by a passerby who found the young orphan on the street. Hugo may have intended it to be a derivative of the French word enfant "child".
The name "Éponine" derives from the ancient Gaul Epponina, wife of Julius Sabinus, who rebelled against the Roman empire. She "became the symbol of great patriotism and virtue" by protecting her husband for many years and by choosing to die with him when he was finally captured. The name was quite common among both Republicans and Bonapartists in post-Revolutionary France. Her sister's name "Azelma" also derives from the name of a loyal wife who dies with her husband, the wife of Abdul-aziz, a north African warrior who fights Napoleon. Hugo explains both names as the product of Mme Thénadier's love of "stupid romances", melodramatic novels on exotic themes with exaggeratedly noble characters. Hugo says such names were typical of the period, when there was "an anarchy of baptismal names" as working-class people increasingly gave their children exotic or grandiose names, while the upper classes intentionally adopted lowly-sounding names.
A real, though not very common, surname from the Haute-Loire region. At first glance it looks like it could be derived from "enjôler," to charm or beguile, but its real root is the Occitan "enjeura," to terrify. A charming youth capable of being terrible...