The boy lives in a beautiful island is also correct. We can question the context for when each one is more appropriate, but as a stand alone phrase and in the context of learing greek, both in and on should be accepted :)
I've never heard "in an island" -- it sounds to me as if someone is physically inside the island, under the ground -- where does one say that?
What about το αγόρι μένει σε ένα όμορφο νησί;
That's correct as well.