Hi Don,
We do this all the time. We spend the last night before a morning flight at the B&B Hotel Fiumicino. There are actually two of these hotels, right across the tracks from each other, at the first train stop before/ after the airport. They are Fiera 1 & Fiera 2 at the Parco Leonardo stop. It is extremely convenient, and they both have a shuttle that runs from early morning to late at night. The rooms are simple, or sparse, but clean and the beds are comfortable. Much cheaper than the Hilton, and taxis from the airport to the residential areas of Fiumicino are expensive. Min. €35.00.
With the train, you can easily get into Rome and wander about and then get back on at Termini, or Ostiense, or Trastevere to go back to the hotel.
It is popular and busy, so book early.
Here is where their shuttle picks up at FCO.