I am one of those who Hate the Termini area. I don't like the "unsavory" types that hang around train stations (it seems) in every area of the world. If I were on a flight that departed at 8am, I'd stay at the airport. But if Termini/Trastevere were the question, for sure I'd choose Trastevere.
I've always spent my night before departure IN Rome, had an early dinner, a last look around, a bit of weeping for having to leave, but on the last trip, visited Rome for 3 nights (4th time) at the beginning of the trip, ended in Sicily, flew to Rome and spent the night before the U.S. at one of the local airport hotels and actually enjoyed the little town. Had a nice walk along the boat docks, and enjoyed a nice early meal and took the early shuttle (15 mins) to the airport for departure..Didn't worry about getting stuck in traffic, etc.