My favorite place to eat in Luzern was Manora. It's on the top floor of the Manor department store in the old town (Manor is a chain, with stores in most Swiss cities). It's a "cafeteria," but much nicer than that term implies, and the food is prepared fresh for you, rather than sitting on steam tablets. For instance, if you get fish, a piece of raw fish is taken off the ice and cooked for you. For 3 CHF, you can add beilage to any main dish, meaning all the starches and veggies you can load onto the plate. It's delicious (the food comes from their ground floor supermarket, which is quite fancy) and half the price of a regular restaurant. The only catches are, it's always crowded (the Swiss know it's the best deal in town), and it's only open department store hours, which in Luzern means early dinner Monday through Friday, lunch only on Saturdays, and no meals on Sundays.
The Tibits referenced in LindaL's post is open 7 days a week for lunch and dinner, and is also good value.
If, like me, you are daunted by the prices of regular restaurants, in addition to the Manora and Tibits, you can get prepared foods from supermarkets (sandwiches, pasta salads, precooked eggs called "picnic eggs," etc). There's a Migros and a Coop supermarket in the old town (closed Sundays), a Coop in the train station (closed Sundays) and an Aperto in the train station (not as nice as the others, but open Sundays).