last day of trip in Pompeii - what airport should we use

Naples would work, but Rome might have better flights and is only 1 hour away from Naples by train. Always stay a night or two in the closest major city to your flight home.

Are you aware of how well-served Pompeii Scavi is by the commuter train (Circumvesuviana) which runs almost every half-hour? If you like the time and price of the air trip from Naples, there is no reason not to spend the night in Naples, and have the hotel pre-book a fixed-price taxi ride to the Naples airport. That said, as in the previous reply, Rome has many more flights to the U.S. We flew (only) into Naples, from London on Easy Jet. Because it arrived at 8:30PM, we spent the night in Naples although our real destination was Sorrento.

It would have been helpful to know where you are sleeping the night before you see Pompeii. Reading between the lines of your post, it may be very far from Pompeii. Did you have any other interest in Naples? I ask, because a night or two in a hotel within walking distance of the Naples central train station might work out well for you. Note that many visitors find Naples too frenzied and urban for their taste. The Naples Archeological Museum was a must for us, since it has the "originals" of many of the frescoes and statues you'll see replicas of in Pompeii. You should allow at least two to four hours after entering the historic site, to see Pompeii.

Bottom line: I don't see any particular need to spend the last night "in Pompeii." There are only one or two hotels in town, otherwise nowhere, and that leaves you further from any airports. If you search this and other forums, you'll find people who "do" Pompeii as a one-day train trip from Rome. But it does take up the whole day. I don't personally recommend it.

There is no theoretical reason why you couldn't see Pompeii, train to Naples, train to Rome, train to an airport hotel if your flight is very early. But most people would rather spend the evening in an interesting city and pay for a cab to the airport if necessary.

Edit: I didn't make it explicit enough that Pompeii Scavi (excavation) is the name of a Circumvesuviana station, very, very far from the modern city of Pompeii, which has a national railroad (i.e. long-distance railroad) station called "Pompeii". But the Circumvesuviana station is right across from the entrance to the ancient site. I also meant that there are only two hotels within walking distance of the ancient site; I suppose there are more in the new city of Pompeii, but that's a business destination, not a tourist destination. And again, it's some distance away.

1-There are no flights from NAP to the US (except for a seasonal weekly pair with Meridiana Fly from/to JFK)
2-To fly from NAP to the US you must connect through a European hub first (e.g. CDG with AirFrance, MUC with Lufthansa), that generally requires a very early flight out of NAP.
3-Pompeii has limited hotel options and not much to do other than the excavations site, which however closes at 07:30pm, therefore you have plenty of time to go back to Naples, where I would stay overnight (the Chiaia area near Piazza Plebiscito is best).
If you don't find suitable flights out of NAP, then Rome will have plenty more flight options. However you need to spend the night before your flight back in Rome. Going to Rome Airport from Naples takes at least 2 hours by train and it's not something you want to do the morning you fly back to America, because things happen and to avoid unnecessary stress it's best to be close to the airport on your last night. Rome city center is only 30-35 min by taxi or train from the airport.

I have stayed a night in Pompei and don't have any reason to recommend it. Either Naples or Sorrento would be more interesting.

Thank you all for your helpful advice. I think I will be staying in Naples or Rome and doing a day trip to Pompei.