Where to stay for Rome quick overnight layover if we are catching cruise ship in civitevecchia

If you want to stay in Rome to have some free time to explore then stay in the area of Termini train station. You can leave your bags at the hotel while exploring and pick them up and walk a short way to the train station. We did this years ago. Once at the port area, we walked it to the ship, but it was a long walk 25 minutes maybe).
There are nicer areas to stay but for one night I chose the Termini area to cut down on the hassle with bags and arranging transport. It is easy to use the train to get to the port. Others will have to help with which train as I don't remember. It isn't hard to do.
You can also connect to Termini via the subway from other parts of the city but I can't give any advice on that!

If you only have one night in Roma, I would spend it in the historic center, near Piazza Navona/Campo de' Fiori/Pantheon.

No need to stay near Termini, two other stations will get you to Citivecchia. Stazione Trastevere, 1hour 5min and Roma S. Pietro. 46 min. Then a 10 min. Taxi to the port.

If it's within your budget, Starhotels Michelangelo Roma is a five minutes walk from the Roma San Pietro train station - and it's right by the Vatican, too :)

Regardless of where you stay in Rome, when you exit the train station at Civitivecchia, simply turn right and walk. It is an easy walk (level or slightly down hill) albeit 15 to 30 minutes depending on your pace. You can go to googlemaps.com and use the street view to "walk" the route before you ever leave home and see if it is walkable for you. Once at the entrance to the cruise port there should be shuttle busses to take you to you ship. If memory serves correctly, a taxi or two were available (but not all that plentiful) at the train station the past several times we were there. Perhaps there are more now.