We will be on a cruise during May with Rome as a stop for the day. We have been to Rome before (on our own as well as a previous cruise) and have seen the major sights. Can anyone recommend another place/town to visit that day.Thanks

I would go back to Rome and see some of the churches/museums/historic sights you have not yet visited. It is hard to make specific recommendations without knowing what exactly you have seen.

Ostia Antica, the Baths of Caracalla and the Appian way are all insteresting if you have never seen them before
There are of course hundreds of churches in Rome, many with beautiful artwork:
http://travel.nytimes.com/2007/04/22/travel/22Rome.html
some sights on my still to do list after several lists to Rome include the Villa Farnesina and Villa Giulia. We have only seen a fraction of the Vatican collections--last time we went to the Pinacoteca and the modern art gallery

There is so much to see. Assuming that your "on our own" stay wasn't at least a month, you have not even scratched the surface.
Even if you just go and wander, it's worth it.

Orvieto is frequently mentioned as a good day trip from Rome. It is accessible by train.

"A stop for the day." How long is that? 8 hours, 10 hours, 24 hours?

Thank you, Big Red, I will read up on Orvieto.

If you are interested in Etruscan history, Cerveteri and Tarquinia are both closer to the port area than Orvieto. I am not sure how transportation options/timing compare.

The Castelli Romani area would be another possibility especially if you rented a car. I believe there is a strawberry festival in Nemi in May