Stay near the Termini station so you can easily take the Leonardo Express to the airport, runs every half hour, takes 32 minutes, costs 14 euros. Termini is also a hub for the Metro and buses for local sightseeing.
I haven't stayed there, but hotels near Termini recommended in the RS guidebook are Residenza Cellini, IQ Hotel, Hotel Aberdeen, Hotel Opera Roma, Hotel Sonya, Target Inn, Hotel Oceania, Hotel Selene Roma, Bellesuite Rome, Hotel Nardizzi Americana, Hotel Italia Roma, Hotel Margaret, Hotel Select Garden, The Beehive, Hotel Sileo, and Hotel Robinson. Also some hostels but as a cruiser I assume you've made the leap to hotels. ;-)
Staying "in between" the center and FCO would be your worst choice, I think, convenient to neither. I'd be interested in where the previous poster stayed in or around Ostia Antica before a morning flight. I tried to figure out those logistics for my trip last month but couldn't make it work. Ostia is a great sight but not a priority for a visit as short as yours will be.