If you stay in Waikiki, you don't even need a bike. You can just walk. If you need to a bit farther, take public transportation. To go even further, e.g. north shore, join a day tour.
One of my favorite vacation spots is Provincetown in Mass. because I don't need a car. Went to Kauai, but I seemed to be driving everywhere. On any Island, are there any small cute beach communities where I can walk and bike everywhere-to the beach, to eat, to shop?