If an Aura is exiled this way, its owner chooses what it will enchant as it returns to the battlefield. An Aura put onto the battlefield this way doesn't target anything (so it could be attached to a permanent with shroud, for example), but the Aura's enchant ability restricts what it can be attached to. If the Aura can't legally be attached to anything, it remains in exile for the rest of the game.