It seems to me that if the proper name of the island contains the word "Island", then we would always use 'on'. At least, this is true of places I've had direct experience with:
- In 1982, I lived on Long Island, which is in the State of New York.
- My family used to spend every summer on Long Beach Island (in New Jersey).
- I spent last weekend on Block Island.
(Block Island is located in the Atlantic Ocean, off the coast of Rhode Island.)
Speaking of 'Rhode Island', that is not actually an island. Rather, it is a state that happens to have 'Island' as part of its name. In this case, I would say "My brother lives in Rhode Island".
I would tend to use "in" with the plural 'islands' in this sort of sentence:
- He has a home somewhere in the Hawaiian islands.
HOWEVER, when it is again narrowed down to a single island, I would go back to 'on':
- He has a home on one of the Hawaiian islands.