I will have a rental car for a week, and I would rather not drive it into Naples. Can anyone suggest a location where I could pay for overnight parking and catch the train into Naples? I will be leaving from Rome, so any good location between Rome and Naples would work for me.

You should tell us WHY you don't want to drive into Naples. If you're worrying about the traffic, then don't rent a car at all. Driving is fun anywhere in Italy for who loves to drive, but if Neapolitan traffic is scaring you, Roman traffic will scare you as well, I promise.
If you're worried about car theft, you can make the whole trip by train as well. You wouldn't go a very long way from Rome towards Naples if you want to completely exclude the risk - in Southern Lazio already (Gaeta, Formia), it tends to be somewhat unsafe for nice cars, and as soon as you're in Campania, well... at least, the car is not yours, right?
But if you are just worried about your trunk being mugged while you're waiting at a red stoplight (i.e. if you're carrying any valuable luggage), then I agree, leave the car outside Naples.
It's been quite many years since I've visited Pozzuoli, and at that time, there was no such thing as a guarded overnight parking garage. But IF there is, in the meantime, this would be a convenient location for going into Naples center by public transport (busses should be available as well, since it's so close to Naples). But Pozzuoli is not between Rome and Naples, rather, it's west of Naples.

Hi D,

Perhaps you don't need a car at all.

What's your itinerary?

yes
I cheat and park at the airport and take the "Alibus" into the city; it is fairly safe and long-term parking is only about 8euros/day.