It has been too many years for me to give specifics, but I can say to look for little Mom & Pop Asian places in strip shopping centers and such. They are the very best deal, often being less expensive than the food would be if you bought it yourself and cooked it. Many do not have liquor licenses, but will allow you to carry in a 6-pack or bottle (double check this at your hotel to make sure the laws haven't changed since I was last there). Thai, Korean, Vietnamese, Indian, "Hawaiian" (which is a mish-mash of all cuisines), Chinese, Japanese, Filipino. They are all available and delicious and inexpensive in these casual, store-front settings.
Also hitting a grocery store deli can unearth some great carry-out that is exotic (plus the PB&J your kids might need) for those meals you don't want to go out for.
The lunch wagons are great as well. Look for "plate lunch" signs. A plate lunch is a full meal on a paper plate, and usually you have a choice of many types of entrees, plus "One or Two Scoop Rice". Again, very heavy Asian influence, with the addition of macaroni salad (do NOT ask me why, but the Korean population LOVES macaroni salad. Actually, they love anything with orange flavor or mayo).
Believe it or not, 'though I wouldn't do it every day, even McD's can be a cultural experience, especially at breakfast.
Saimin houses are great as well. Again, usually in strip malls. Saimin is basically what we call ramen noodles, but many variations of ingredients and condiments.
A lot of people criticise Oahu and Waikiki, but I've had many wonderful times there. Avoid whatever is being touted as the "latest, newest, trendy" restaurants at all costs. They are fun for a romantic dinner for two, but very $$$$$ and not really great with kids.
Somewhere here we used to have a correspondent who lives in Hawaii. HELP.