grandparent or 1/4 Polynesian for initial eligibility. Otherwise, because ethnicity is a complicated question, the candidate must appeal for consideration at which point it would be at the committee's discretion (it's their award, they can choose how to award it).
I know as a half Asian that it would bother me if someone with a very tenuous claim to being Asian (say, less than 1/8) we eligible for Asian player of the year. But on the other hand, as a half Asian who looks more southern European than anything, I would be annoyed about people questioning my Asianness.