Re: New York City hotel/lodging at a decent price?
Lodging in NYC is pricey . . . good eats are not.
Gray's Papaya or, if you're visiting the Upper East Side (the Met & Guggenheim), Papaya King.
A bazallion pizza places where you can get a huge slice that hangs off the paper plate with a soda for a few bucks. The best New York pizza is thin, with a crispy crust. If you can pick it up by the crust with one hand and the rest of it doesn't flop down, you know you've got the real thing. Cheap, hot, and good.
Chinatown is adjacent to Little Italy and each neighborhood serves up the best of their respective cultures. Eat Chinese for the entree and Italian pastry and coffee for the best value.
One of my favorite "musts" of New York eating is to have a drink, maybe an appetizer, at the Met. You can sit surrounded by priceless sculptures inside or enjoy an incomparable city and Central Park view on the roof. Expensive and . . . come to think about it . . . probably more enjoyable for adults than kids. Oh well, one of my favorites, anyway.
Try to avoid tourist traps. Use your nose. Don't be afraid to walk out, if it smells bad or feels weird there's probably a place right around the corner with a meal of a lifetime.