You can see some splendid things in Rome even if you are just there for a day and a half.
http://www.italymagazine.com/feature...one-night-rome
http://travel.nytimes.com/2010/10/10...anted=all&_r=0
I agree, definitely don't stay at the airport. There's nothing out by FCO. If you've never been to Rome you'll be missing a lot. If convenience not price is your major consideration, I'd just take a cab. The fixed taxi rate is now up to E48. In my opinion the Leonardo Express can be a little challenging if you don't speak any Italian.
I second the opinion that staying near the Termini is not a good idea, especially for a short stay. It's not a nice part of town, and it's over a mile walk to any of the main sites that you'll be wanting to visit. At least the Port Authority in NYC is centrally located just off of Times Square.
For a short trip I'd recommend a hotel as close to the Pantheon as possible. This would put you within a 3-6 minute walk to most of the main sites you'd want to see, e.g., Pantheon itself, Fountain of Trevi, Piazza Navona, Spanish Steps, and 20 minutes to the Coliseum and Forum. The only long walk, but doable, would be to the Vatican.
On a short trip I'd lean more towards sightseeing than going to places that require a ticket.
I was in Rome this past Saturday arrived at 4 PM, with a flight out at 10 AM the next day, so I just had an evening. My stroll went from the Pantheon, to Piazza Navona, Piazza di Popolo, Spanish Steps, a great dinner in Monti, Piazza Venezia, Coliseum, Forum, then back to the hotel. I would recommend those as the sites to visit during a short stay. BTW, I think it's worth seeing the Coliseum and Forum both during the day and at night. Very different perspectives.