What an exciting 16th birthday present. You neglect to say what kind of food your wife and daughter like, and what "budget" means. But they're likely to find everything in Manhattan expensive, so I'll interpret budget to mean as cheap as possible. New York food can be an adventure in the world's cuisines; contemporary NYC's take on the world's cuisines (think Momofuku); and historic NYC classics. I'm guessing the first and the last option are more to your family's needs on this trip.
If they like Thai food, there are very nice, cheaper places in Hell's Kitchen/Theatre District. The best is either Pam's Real Thai (49th and 9th) or Wondee Siam (there are 3 locations near 9th Ave).
If they like Japanese food, they might enjoy trying "ramen" -- this can be had in this part of the city as well, at Sapporo (49th and 7th), or the real happening outlet would be Ramen Totto (just profiled on CBS Sunday Morning) on 52nd and 9th.
Certainly no trip to NYC would be complete without an experience in a Jewish Deli. The best is the 2nd Avenue Deli (33rd and Lex, confusingly). But the Carnegie Deli is also not bad, if they are reluctant to leave the tourist cradle of midtown.
Also, how could one visit NYC and not eat at the Grand Central Oyster Bar? It's wonderful, and a very New York experience, and of course, you'll want to see Grand Central anyways. The oysters are actually reasonably priced -- and if they don't like oysters, the chowder and She-Crab soup are divine -- the menu is huge.
Pizza-wise, I always take out of town visitors on a trek across (that would be walking) the Brooklyn Bridge to eat pizza at Grimaldi's. I think it's the absolute best pizza (outside of New Haven, CT), but be prepared to wait in line outside (an hour or so). Another option for real NY-style pizza would be Arturo's on Houston (remember, it's pronounced "how-ston"). John's would probably suffice, if time or sense of adventure is lacking. The key is to avoid the slice thing -- the places that serve real NY pizza do not tend to sell by the slice.
For the budget eater, Chinatown is a must. (They will probably want to seek knockoff bags and such anyways, right?) I would recommend Nice Green Bo (formerly New Green Bo) for really great Shanghai style cooking, on Bayard St. If they like Vietnamese Pho, that can be a wonderfully cheap meal, and plenty of places in Chinatown for that.
<message edited by bristaylor on Thu, 01/13/11 12:58 AM>