Last night of Rome accommodation

If you will have already “done” Rome, I would suggest the Trastevere area. Much quieter and laid back than the city. We had a lovely stay at the Santa Maria.

I will not have done Rome at all, but I will only have the evening and early morning . Some say to stay at a the airport but really don’t want to waste time at an airport.

Your post says you will have already spent a week in Rome?
Confusing

Trastevere is a great suggestion

If you don’t want to venture far then consider Ostia/Fiumicino beach near FCO

I will have already spent a week in Rome and I will have one more night (after travelling around).

The 1st line of your post, seems to indicate you will 'have done Rome', prior to this stopover.

When in 'the afternoon' are you scheduled to arrive and where does the '230 or so' departure land? If it lands outside the Schengen zone the common wisdom is to be at the airport 3 hrs ahead.

I fly into Rome FCO in the afternoon

From where and when between Noon and 559PM is the scheduled arrival?
An arrival from Athens at 1PM is a much different thing than a 550PM arrival from Istanbul.

Making a useful suggestion without knowing the time frame you will have available is all but impossible

I like the idea of spending more time in Trastevere. I will arrive in Rome mid afternoon and fly out late afternoon the next day to North America..so I am aware that my time is limited. Any good stays in Trastevere that anyone can recommend?

See Trastevere hotel recommendation above