I'm sure I have left a few out, one or two maybe intentionally, but here is a good starter list. I don't think you are going to find really great food 24/7 on Oahu, but some of it isn't bad at all. So... in alphabetical order:
Bob's Big Boy: In Mapunapuna, this is a throwback to the late 60's roadside diner. The food is passable, not great, not awful... ok, a few items might hit either extreme. Stick to the simple stuff, burgers, hotcakes, etc.
Byrons Drive-in: Near the Airport, probably the most varied menu of any of the 24 hour places. This is take-out plate lunch style food, while there is no actual dining room there is a covered lanai with seating. You aren't here for the atmosphere. The food can be surprisingly good, especially the daily specials.
Denny's: In Waikiki on Kuhio. Denny's hardly needs explanation, the quisesential american diner.
Flamingo Pearl City: A "local style" Denny's. There used to be several of these around the island, as far as I know the one in Pearl City is the last. Food is OK, about what you expect for a 24 hour diner.
Jack-in-the-Box: What can I say. They are open 24 hours and they are all over the island... aside from getting something microwaved at 7/11 many times JITB is the only choice in the middle of the night.
Likelike Drive-In: Rebuilt in the late 80's, the menu is pretty much the same as always, except it is a restaurant rather than a drive in. The food is above average for a 24 hour diner, while the saimin and the fried rice have been midnight favorites for the last how many decades.
Liliha Bakery: Closed on Tuesdays, but open 24 hours the other 6 days of the week. Go get the pancakes (what a value) while sitting at the counter (the only seating) and watching the short order cook at work. Don't forget to pick up a package of coco-puffs on your way out.
Wailana Coffee Shop: The oldest 24 hour place in Waikiki, and still one of the few. Fried Chicken and breakfasts are favorites for the late night crowd at Wailana's.
Zippy's: Some love it, some hate it. All in all one of the better places to go when the hunger pangs hit in the middle of the night. Locations around the island. The saimin is pretty good, the burgers arent bad, but plate lunches are really the foundation of the Zippy's empire. Avoid the chicken katsu and all will be OK. I've always liked the clubhouse sandwich (ok, so I'm haole).
Eggs & Things: I am putting this last because it is not open 24 hours, but it is open all night. Great omelet's, used to have great pancakes (last two times I was disappointed), but arguably the best late-nite choice in Waikiki.