Grottaferrota or Fuiggi
There are great options in Grottaferrata (Il Fico Vecchio, Il Tuscolo, La Briciola di Adriana) and in neighboring Frascati (Cacciani, Zaraza, Sapori di Casa).
Not far from Fiuggi in Acuto is Le Colline Ciociare, a superb place.
On balance, I would stay in Frascati, which is a charming town, and eat there or in Grotaferrata.
re: Jeremy M
I think that's a fair assessment. Fiuggi is famous for its mineral water, prescribed for kidney stones. Le Colline Ciociare is a truly wonderful restaurant, in the Michelin star category. In Grottaferrata, we love La Briciola di Adriana, much homier, but not your standard trat. The others you mention are good too. Grottaferrata is closer to Rome than Fiuggi.