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 :)
Because we're using the indefinite article here and not the definite article.
If we used στο (νησί is neuter), we would mean 'on the beautiful island'. Essentially, we have a specific island in mind that we're referring to, and the boy lives on it. We don't make explicit reference to which particular island it is, but it's implied that the people speaking and/or hearing the sentence know which particular island is being referred to.
By using σε ένα, we're saying that the boy lives 'on a beautiful island'. Which particular island it is isn't specified, nor implied, and it may not be known by either the speaker or the receiver - it could be any beautiful island, anywhere. All that is known is that there is a beautiful island somewhere, and that the boy lives on it.
Hope that helps!