Excerpts from Wikipedia's article on naming conventions for planetary moons:
In 1847 the seven then known moons of Saturn were named by John Herschel. Herschel named Saturn's two innermost moons (Mimas and Enceladus) after the mythological Greek Giants, and the outer five after the Titans (Titan, Iapetus) and Titanesses (Tethys, Dione, Rhea) of the same mythology.... Since the outer moons fall naturally into three groups, one group is named after Norse giants, one after Gallic giants, and one after Inuit giants....In addition, individual geological features on moons have their own naming conventions: those of Ariel are named for spirits of light, those of Umbriel bear the names of darker spirits, and those of Miranda continue the Shakespearean theme of Uranian satellites. Features on Titan get their names from such varied concepts as "sacred or enchanted places", "planets from the fictional Dune universe created by Frank Herbert", and "islands on Earth that are not politically independent".
[Uranus:] Herschel, instead of assigning names from Greek mythology, named the moons after magical spirits in English literature: the fairies Oberon and Titania from William Shakespeare's A Midsummer Night's Dream, and the sylphs Ariel and Umbriel from Alexander Pope's The Rape of the Lock (Ariel is also a sprite in Shakespeare's The Tempest). The reasoning was presumably that Uranus, as god of the sky and air, would be attended by spirits of the air.
Subsequent names, rather than continuing the "airy spirits" theme (only Puck and Mab continuing the trend), have focused on Herschel's source material.... Current IAU practice is to name moons after characters from Shakespeare's plays and The Rape of the Lock.