Bolsena's restaurants cluster along the lakefront and in the old town, and the local speciality is eel — cooked alla brace over charcoal or in umido with tomato and herbs. The eel of Bolsena has been celebrated (and lamented) in literature for centuries. If you are open to trying it, this is the place; several lakefront restaurants serve it well in summer.
Otherwise, the cooking is straightforward central Italian — pasta with local pork sauces, grilled meat, and in summer good lake fish. The town is small enough that almost any restaurant on the main square or the lakefront will serve a decent meal.