I lived on Oahu, so I'm partial to that island. Depending on the amount of money you're willing to spend, you can "island hop" to another island for a day and take tours while you're there to see all of the highlights.
There are tons of things to do on Oahu (just as the other islands, I'm sure) that don't cost a lot, or any, money. Tour the old bunkers in Diamond Head, go to the Pali Look-out, spend some time on Waikiki Beach to say you've "been there," although there are better beaches on the island. Watch the torches being lit every evening at sunset in downtown Honolulu. Drive down the Kalani'anaole Highway for breathtaking views and photo opportunities. Hike through the forest to see Manoa Falls...go to the North Shore to see the famous waves (especially if you go during the late winter or early spring). You can also go whale watching along the coast during the same time -- all of this completely free! Lots of paid attractions worth seeing...go snorkeling with the tropical fish in Hanauma Bay, see the Dole Pineapple Plantation, Pearl Harbor and the USS Arizona...go to the Aloha Stadium on a Saturday for an amazing flea market...The Polynesian Cultural Center...and my favorite place for a luau is Paradise Cove.
Wow! Can you tell I miss Hawaii? I'm not sure where you should stay (I only stayed in one hotel while I lived there), but I would recommend Oahu if you can only see one island.
Although...on the Big Island you could see an active volcano, snow ski, and tan at the beach all in the same day....
:o)