There is no dense restaurant district in Carthage proper — the archaeological zones are residential or open, and most visitors eat in the cafés around the TGM stations or up the hill in Sidi Bou Said. The Café des Délices on the Sidi Bou Said cliff, perched halfway down toward the marina, has the best views and reasonable mint tea with pine nuts. Au Bon Vieux Temps on the same cliff is a slightly fancier sit-down option, mostly French-Tunisian fusion.
For a proper meal, the restaurants along the seafront in La Marsa (one TGM stop north) are the best in the area: the Slovenia (Tunisian-Italian) is popular and L'Antica is good for pizza. Down at the bottom of the Sidi Bou Said cliff, the Marina has several seafood places that fill up with weekenders from Tunis.
The Antonine Baths neighbourhood has a couple of small cafés but is otherwise quiet — bring water and snacks if you plan a long day among the ruins.