We are staying at Hotel Barberini. Would anyone have recommendations for a local restaurant nearby?
Not too far away (10 minutes on foot) is Osteria 44. Molto buono! I also like Colline Emiliane. Reservations recommended at both.
Pizzeria Ai Forni Della Soffitta is less then 20 minutes on foot and quite good. If you make a reservation you get a better table. My favorite gelateria in all of Rome is also nearby, Come il Latte.
If you just walk around, you'll see quite a few. Avoid the ones that have food displayed in the window or someone outside touting the place - those are for tourists. Locals avoid them.
We enjoyed Hostaria Romana a couple of years ago. Not far from Piazza Barberini. Although we didn't have reservations, we lucked out and were able to get a table for dinner. It is a small place, so I would recommend making a reservation. The food was excellent. Here is a link to their website: http://www.hostariaromana.it/
Buon Viaggio!