Here follow my favorites (everyone has their own)
Gigetto: This is usually my go-to place in the Ghetto. It's the oldest restaurant in the neighborhood, and serves perfectly made carciofi alla giudea as well as all the other Roman favorites. Since it is not Kosher, you can also get carbonara, gricia, amatriciana. All wonderful. If you go in the evening, ask for a table outside. In the summer, it's too hot at lunch time, so sit inside. There are tourists here, but just as many regulars.
Sora Margherita: A true hole in the wall. Although the prices have gone up a bit, it is still exactly as it has been for years. Simple paper topped tables, hand written menu that changes daily. Their speciality is caccio e pepe with ricotta.
Pompiere: Right across the street from Sora Margherita, this classic Roman restaurant often gets overlooked. It's on the 2nd floor, and so the rooms are large, airy and some are frescoed. It's a bit pricier than the first two, but still not bad. Go here for both Jewish and Roman classics, as well standards like saltimboca, which they do very well. I love their wild cherry crostata. Almost never any tourists.
Il Giardino Romano This is my Monday back up restaurant. Opened by an ex waiter from Gigetto, it's pretty much the same menu and cooking. Extremely nice service, and I love their little courtyard in the back. Excellent fried artichokes. For dessert, although it's not Roman, ask for a plate of Egyotian pastries. The owner is Egyptian and they are the best I"ve had in Rome.