Andrew Sullivan quotes a Christopher Hitchens article in Slate that claims we should name Mumbai Bombay because Mumbai is a new name "forcibly" imposed by Hindu fundamentalists based on the name of a Hindu goddess. But as Larison remarks, the name Mumbai has been in use in the region for centuries by speakers of Marathi (and a slightly different prononciation in Gujarati, although the Hindi prononciation (Bambai) is closer to the former name). The prononciation Bombay comes from an anglicized version of the Portugese version of the name of the place; it is by no mean the "original" name of the place but merely an approximation made by eurocentric European merchants and soldiers.
But even if the name had been imposed by Hindu fundamentalists for religious reasons, I do not see why I should favor the colonial name over it. While I dislike religious fundamentalism as much as Hitchens, I have trouble seeing how a colonial name is inherently superior over a religious based name.