Cheap and Hawaii do not go together. Hawaii is pricey compared to the rest of the US.

I would recommend one airline for both sectors UK-LAX/SFO/EWR-Hawaii; I am sure you will find it is cheaper and there is easy of luggage transfer (if only back to the UK).

Depending on what you want to get out of your trip, I would suggest the big island of Hawaii itself; only because there are more options of things to do there. Oahu is too built up if you want scenic beauty. Maui is lovely but pricey and more up-market. If you want to see an actually live volcano with lava flows - Hawaii has a one or two.

All I can say is that prior to going, I really wanted to see Hawaii. Having been there, my feedback is "waste for time and money" I can get the same for a fraction of the price in Thailand or Bali.

The Hawaiian island are like the Canaries - dry beaches on the southern part of the island and lush green forests on the wet northern side of the island.

It is a very long flight for the children. Just remember that Hawaii is almost on the other side of the world - 10 hours time difference during UK winters and 11 hours behind in summertime