It depends on what you're looking for. Oahu has all the amenities since Honolulu is a pretty big city, but there's still plenty of beautiful beaches and rural places elsewhere on the island. There's also a small but blossoming tech scene in Honolulu. If you want complete peace and quiet, then I would suggest the Big Island or Maui. The Big Island is like being in the middle of the Everglades in Florida, with a few small towns and nothing for miles. Maui is a little more suburban and cosmopolitan, but still somewhat rural since there's no big city like Oahu. While Kauai is absolutely beautiful, it's very small and far more touristy and commercialized — I'm not a big fan.

Renting out residential homes is technically illegal, at least on Oahu, but people still do it and you can still probably find tons of people renting out rooms in their homes for short-term stay on all the islands on Craigslist. On Oahu there are a lot of short-term accommodations in Waikiki, but Waikiki is very touristy, somewhat expensive, and most of the places are converted hotel rooms with no kitchens. A few centrally-located neighborhoods you might want to check that are relatively affordable are Manoa, Palolo, Kaimuki, and Makiki. Kailua and Kaneohe are also nice, but they're on the other side of the island and they're inconvenient without a car.