Do you like chicken? Do you like waffles? Dame's Chicken and Waffles is great, but get a reservation a day in advance.
Pompieri Pizza is also good in downtown Durham. Rue Cler is good for relatively inexpensive French cuisine. For fine dining, Nana's is great. La Residence in Chapel Hill is great.
Ninth Street in Durham and parts of Franklin Street in Chapel Hill are good. Spanky's on Franklin is pretty good and is mentioned in a Scharpling and Wurster call, although that location is in Newbridge. Tom loves the Italian Pizzeria III on West Franklin.
If you're a hipster or love hipsters, I'd recommend The Federal. If not, not. I like Alivia's nearby. Those are very near James Joyce.