In The Tempest, who is the rightful owner of the island?

That's a very good question! The short answer is: it's complicated. Prospero, like the proto-imperialist he is, thinks the island belongs to him. After all, he was the one who conquered the place, established some measure of law and order, and for good measure drove off the evil witch Sycorax, thus freeing Ariel from his captivity inside a tree. If the island doesn't belong to Prospero, then to whom does it belong?
Caliban's answer would be "me." Caliban resents the fact that Prospero's taken over his island home and has proceeded to treat him like a slave. Over time, he becomes more and more resentful of his master, to the extent that he urges two members of the shipwrecked party—Stefano and Trinculo—to kill Prospero, which Caliban believes will restore him to his rightful position on the island.
Ultimately, Prospero seems to realize that the island doesn't really belong to him after all. This doesn't necessarily mean that he's become convinced of the justness of Caliban's cause, it's just that he now has a chance to return to his beloved Milan, where he will once more reign as Duke. For now, however, it seems that ownership of the island will pass to Caliban. At least, that is, until the next deposed ruler appears.