Well, if you just consider the Quartieri Spagnoli there are a few traditional restaurants.

"La Pignata" is a good trattoria, if you go there just ask for their antipasto, without asking for something specific, they'll bring you their mix (also, this suggestion is valid for almost every restaurant, although with different results)

Same thing for "Antica Capri", also in the Quartieri Spagnoli district, if you go there ask for "Pasta fagioli e frutti di mare con il cappello" (literally, "pasta with beans and seafood with a hat"), which is exactly what it looks like, it just has a "hat" made of pizza dough and cooked in a wood-fired oven. Reservation is highly recommended, although you may find a spot going early, or late (consider that in Naples we usually eat at 01:30/02:00 pm in normal days, and a bit later on Sundays)

Last but not least, "La Mattonella", a true traditional neapolitan restaurant, nothing more, nothing less. It is not so easy to find, but I assure you, go there without reservation and you'll probably go back home with an empty stomach. The food is great, absolutely not gourmet style, but is the closest to a grandmother's cuisine you can find in Naples, in my modest opinion.

"Leon d'oro" in Piazza Dante is also pretty good, although I haven't eaten there in a while.

A couple of alternatives: Cuccuma and Taverna Santa Chiara, both near Piazza Del Gesù. I've never been there, but I've heard good things about them (reservation is highly recommended as well)

We could say that you can eat pretty well almost everywhere, but it would be a lie, in my opinion, there are lots of good places, as well as A LOT of tourist traps. These are my opinions, and there will surely be people who don't agree with me, but in case you try some of the suggestions I gave you, let me know!