We are taking a train from Florence 2/28/17 to Rome in order to have easy
morning access for a 10:35am flight out of FCO 3/1/17. Where can we spend the
night so it's easy (shuttles?) to get to the airport? We like modest not 4-star lodging
There are frequent Leonardo Express trains to the airport from Roma Termini station, and there are slightly slower but cheaper regionale from Roma Tiburtina. The Trastevere hotels can be very convenient to Tiburtina, or an short taxi (or tram) ride away.
At 10.30 is it a direct flight to the US or a connecting flight to somewhere else in Europe and then back to the US? Small difference in recommendations.
This has a shuttle to an from airport FCO and it is currently $86
http://booking.com/2ae488bd3ba0
I'd stay around Rome Termini or a couple of underground stops away so you can get on the Leonardo Express Train promptly. You should be okay.
You could also get one of the shuttles to pick you up, but it'll be more $.
I'm just glad you didn't take a 6:00-7:00 a.m. flight out of FCO. They all go to another European gateway city where you'll be sitting until mid day's flight out. Flying long distance is hard enough--without having to start on 2-3 hours' sleep in order to make that 6:00 a.m. flight.