Van Rental Rome

I myself would not want to drive in Rome, with the traffic as well as trying to figure out the ZTL zones. Have you looked at taking the train? There are quite a few direct trains every day.

If all you need it for is for the day trip to Tivoli, why don't you all take the bus or the train?
For Villa Adriana, which is a bit outside of town, you have 3 choices.
1. Take the Metro to Ponte Mammolo, then take the COTRAL bus to Villa Adriana.
2. Take the train from Roma Tiburtina to Tivoli, then take the bus from Tivoli station to the Villa Adriana.
3. Take the train to Bagni di Tivoli, then the Trenitalia bus link to Villa Adriana.

If you are interested in Villa D'Este, that one is located in Tivoli town itself. From the station it's a 20 min walk, so a train would do. But if you don't want to walk, take the bus for a couple of stops, and get off near Villa d'Este.

A 7 passenger van will cost you at least $200 a day to rent, plus gas and tolls, which however is not much given the short distance. The driving time is also at least one hour each way, to which you need to add the time necessary for going to the rental agency and for the paperwork. But the biggest issue is that driving in Rome is also very chaotic, and you must know where you are going or you might accidentally enter the ZTL or drive on a bus lane only, both of which will result in automatic fines issued by plenty of traffic cameras.

one other consideration for renting a car or people carrier for that trip is that each driver who doesn't have an EU country (+EEA + UK) driving license requires an IDP. That's available in the USA and Canada at AAA/CAA offices for around $20 plus photo.