I like to stay in Livingston which is a great old railway town around 50 miles North of the North Entrance to Yellowstone and there are plenty of restaurants there. On the budget side there is the Pop Stand in Paradise Valley which is the valley you will drive in between Livingston and the North Entrance (great burgers, etc and the local Wilcoxson ice cream), there’s Mark’s Drive-in (similar), there is the Sport, plus there are the usual chains. Gardner which is at the North Entrance has some decent places also although not as wide a variety as Livingston. Cody is good but kinda far, West Yellowstone is okay although convenient.