If I remember correctly you are going in December.
If you do decide on flying to another island I would consider Kauai for Waimea Canyon (the Grand Canyon of the Pacific.)
@mjm1 I'd expect Waimea Canyon will be socked in during December.
OP, for all the reasons mentioned above, I'd stay right there on Oahu. Drive around and see some things. Tantalus Drive, for example, has remarkable views over Honolulu and Diamond Head. The Pali Lookout, as mentioned above, provides incredible views of the windward side of Oahu. Kailua Beach, Lanikai, Waimanalo, Makapu'u Beach, and even down to Sandy Beach on the way to Hanauma Bay are all great stops. Going the other direction from the Pali, Kaneohe Bay to up past Chinaman's Hat offers some amazing coastline, great swimming, and easy fun. Swim a bit, have a progressive picnic, take the Jurassic Park tours at Kualoa Ranch. Stopping at the surf beaches on the way up to the North Shore can be a great time, especially in December, when the surf is higher, weather is rougher, it's windier, and often rainier. If the weather is dry, and swimming possible, jumping off the Rock in Waimea Bay is exhilarating. Your boys are certain to enjoy themselves. Putzing around Haleiwa Town is always fun, and there are some excellent shops, galleries, and restaurants there. Keep coming back around the center of the island, and make a stop at Dole Pineapple. Yes, it's a tourist trap, but who doesn't enjoy a fresh Dole Pineapple Whip?
There is so much to do on Oahu that many tourists never see. Go to the Bishop Museum, and learn about the amazing history of Hawaii, where the Island people came from, and how their society developed. See Iolani Palace, and learn how the Monarchy was stolen from the Hawaiian Royalty. Visit one of the incredible Botanical Gardens, and learn about the amazing plants that are found in the Islands.
There is a ton more to see and do on Oahu, but you only have one day. Or do you? You may want to extend your vacation, or make it even more adventurous that you're already planning.
Above all, have fun. Just because you sold your timeshare doesn't mean you won't be back.
Good luck, and Aloha!
Dave