Looking for suggestions:
Arrive in port at Civitavecchia on Saturday early morning and fly out of FCO at 6:20 in the morning on Sunday. Wanting to stay in Rome over night but not sure if this would work on logistically. Any recommendations on how to swing this or would it just make more sense to stay closer to the airport over night? I just don't want to waste a night in Rome. If Rome, what area would be best knowing we need to catch an early cab or train on Sunday. thanks

Assuming you're flying internationally (to the U.S.), you need to be at the airport 3 hours in advance. I can't imagine getting to FCO from central Rome at that time of the morning would be very easy to do, unless there's a car service or something you can reserve with in advance. Why not go into Rome, ditch your luggage and spend the day and early evening enjoying the sights before heading to an airport location to sleep for a few hours? Or, be a baller and stash your bags and pull an all nighter, which still leaves you in the position of getting to FCO in the middle of the night.

There are indeed car services you can reserve, and traffic will be minimal at that awfully early hour. If you want the day in Rome, take the day and overnight in Rome.

Alternatively, you could spend a relaxing day in Ostia with its seafront promenade and Ostia Antica ruins. Very short ride to the airport from there, but you’d still have to reserve early morning transport.

I would stay in Rome overnight and have your hotel pre book a taxi for the morning or book a car service. If you check in online the day before you will just have to do bag drop and security- if you are American there is a dedicated US line through security at Rome's airport. Or there was when we had an early flight out last June. Not being American we didn't use it but it is there and very convenient if you are.
We stayed in Monti and it was zero problem getting to and from the airport. Seamless and fast especially that early morning flight.
Up to you of course but we found getting there 3 hours prior for our early flight was a bit of a waste of time as the gate agents didn't turn up for ages and ages to do bag drop, then they took forever to get their acts together and we had been standing (or sitting or lying on the floor in some cases lol) for about 2 hours.

Airport security doesn't open until 4.30. Unless it's changed in the last couple of months.

You're going to need a cab or car service since the trains stop I think before midnight. There are some buses but that would mean an even earlier start.

Problem is unless you stay at the airport hotel staying close doesn't help much. You'll still need a cab. The airport hotel you can just walk to the terminal.

You said, “Looking for suggestions.” I know you meant about where to stay, etc. but my suggestion (though it may be impossible at this point, or impractical) is try to change to a later flight. Otherwise, a hotel the night before is almost a waste.

I would stay in Rome. I've been in this situation and have tried all the possible solutions.

The airport hotel is indeed right at the airport, but it's a big airport and it's a long walk. Plus an airport is a big bore compared to Rome. We tried staying there once, but I wouldn't do it again.

If you stay in Ostia, you could visit Ostia Antica, the ancient Roman port city, a spectacular archeologic site, bigger than Pompeii. If this would interest you, I suggest you stay in a hotel or B&B within walking distance of the site. You would still have to arrange transport for the morning, and it wouldn't be much cheaper or more convenient than if you stayed in Rome.

It doesn't matter where in Rome you stay. I would decide on something you want to do in your limited time, and stay in a hotel that's relatively convenient for that activity.

There are several reliable car services that cost about the same at a taxi, and will help with your luggage. One is Romecabs.com and the other is romeshuttlelimousine.com. Both also offer service from Civitavecchia to hotels in Rome.

How far into the future is this cruise and single night? Unless it's very soon, your departing flight could have a schedule change (or two) between now and then. I'd still follow the advice to stay IN Rome regardless of the departure schedule, but the timing might end up being more comfortable than it appears today.