hotel by the airport in Rome

Try the Hilton Garden Inn Rome Airport for rooms at around $115.
FCO airport is famous for those 6:00 a.m. flights where you just fly to another European gateway airport until mid day. I prefer just not to schedule any flight before 9:00 a.m. It's hard enough flying 10-12 hours when you have little to no sleep the night before knowing you have to be at the airport so early.
We prefer to hire a car to meet us at our hotel/B&B to take us to FCO--if just so we don't have to haul luggage into Termini and onto the Leonardo Express train to the airport.

I stayed at the Hilton. An easy walk to the terminal.

We stayed at the Hilton Garden Inn several years ago. The hotel is fine, but there are no other restaurants in the neighborhood. And it is probably not in walking distance to the airport. Patricia stayed in the Hilton which is on airport property. If I was doing it again and the hotel had a shuttle, I'd find something in the area just a little further east and on the other side of the autostrada. On maps there seem to be more restaurants, hotels, and a market available there. Have a great trip.

Ina,

Depending on what time your flight departs, the easiest solution would be the Rome Airport Hilton, which is within easy walking distance to the terminals via a covered walkway. If your flight doesn't depart until late in the afternoon, you can easily stay in Rome the last night, and enjoy a fine meal to celebrate the end of a wonderful Italian holiday. If you decide to do that, take the Leonardo Express to the airport (32 minutes, €14 PP).

We stayed at the Hotel Tiber in 2014 for our last night. Clean, modern hotel with a shuttle to the airport for a small fee. They have a decent restaurant and offer breakfast included with certain room rates. We took the Leonardo Express train from Rome to the airport and then got the shuttle to the hotel to get there the night before.

Hotel Hilton. Easy walk to the airport. Hotel also has a complimentary shuttle bus to Downtown Rome. The bus is on schedule.

Note that the posts above are referring to two different Hiltons.

The Rome Airport Hilton is actually connected to the terminal via walkways. The Hilton Garden Inn Rome Airport is much cheaper, but it's not connected to the terminals (you need to take a shuttle bus).

What time is your flight home? Many find it just as easy to take a taxi from central Rome to the airport. A taxi to the airport has a fixed rate of €48, or you can pre-book a car for a similar rate (companies such as Rome Cabs and Rome in Limo are often recommended here - no personal experience). If your flight is early, a taxi will only take about 35 minutes, since you're going earlier than rush hour and going in the opposite direction.

We have stayed at the Rome FCO Hilton several times. There are several places for dinner in the hotel and it is an easy walk from rental car drop off and to the terminals.