Vatican City, Rome inexpensive places to eat
Here are some ideas which have been useful to us on many trips. Last time we were at these places was 3 years ago, but most stay the same over years. We will be back there in December.
These are quite close to Vatican City or in it - not long walks to get to from St. Peter Piazza
1. Giarrosto Dal Toscano at Via Germanica 56, near the Vatican Museum. Huge fireplace/grill for steaks, other meats, and slabs of meat on hooks. Try the Pasta Giulio Caesaro or pasta with boar sauce (cinghale) or rabbit/hare sauce (coniglie). Great grilled meats. Try for lunch when you are near the Vatican Museum.
2. At the corner next to Giarrosto Dal Toscano, on Via Ottaviano is a combination grocery store, candy shop, and café (not sure of the name) where you can browse for food items to bring home (olive oil, porcini mushrooms, Italian candies, etc.).
3. Nearer to St. Peters just off the Borgo Pio is Hostaria Orfeo da Cesaretta at Vicolo dOrfeo 20. Small neighborhood restaurant with a tourist menu. A delicious dish they serve is risotto alla crema di spinaci (creamed spinach risotto). Also serve spaghetti alla carbonara, a traditional pasta dish made with eggs, bacon, and cream. Good choice for lunch.
One other recommendation I would make is to get a copy of ACCESS: Rome (the guidebook) and try the restaurants they list. You can assess price levels and food quality by their ratings. Many of the places we have eaten at and enjoyed in Rome we found in that book. Same goes for their book on Florence and Venice.