I used to live there, and I can tell you the places that were there when I lived there, and hopefully they'restill open! :)
Banana Boat-Very inexpensive, burgers, fish, conch (pronounced CONK), etc.
Hey Jose- a mexican themed restaurant that really serves everything from pizza to burgers.
Top O' the Cove Deli- I know the people who run the place, they're good for cheap sandwhich style lunches and pastry breakfasts. (They are off the main road, connected to NAPA auto parts.
There was a place called "Dora's"- a very local restarant good only for Take-out, but about as authentic of an island meal as you will find, and EXTREMELY cheap.
Stay away from: The Gecko Grill, Alfred's, and any place in Grace Bay if you want cheap prices! :)
Lastly, I know it is expensive, but when I lived there, $30 bucks per person will buy you admission to the buffet dinner at Club Med, which includes dessert and drinks, and you can stay for the show afterwards. They usually had a theme for the meals (Italian, Mexican, etc) and were very good.
Hope that helps.