It's more than that, as other tropical islands do not have the same amount of rainbows, though they have the rain and sunshine happening.

It is that the islands are new (Big Island is still growing) and so the mountains are still high enough to make all the rain fall, at increasing altitude all the way up the mountains. So pretty much all the time rain is falling somewhere on the windward sides but it is up the mountain so sea level still has clear skies so the sun can do the light bouncing around in rain drops thing to make rainbows.

On the windward side, there are mountains and rain and waterfalls and rainbows. On the leeward side there are near desert conditions and few rainbows. Hawaii has a couple of places with over 400 inches of rain a year, with basically the most rain outside of places that have seasonal monsoons.