Thanks to everyone for expanding my knowledge of good Roman food! I have recently tried Antico Arco and Sanpietrino based on your recommendations, and I was so pleased. They really blew other places I have eaten out of the water.
I have been wanting to try a good Gnocchi alla Romana, because I have seen them fresh/uncooked at the pasta al'uovo place near my house. When I see gnocchi on the Thursday menus, I don't often see gnocchi alla romana, but maybe I am not looking at the right places! I guess I should just try and make them myself because they look pretty easy to make.
I would be happy to hear recommendations for any kind of gnocchi in Rome as my boyfriend is crazy about them.