I recently rented a car in Rome, and I think I found a way that saved me time and money. This was with Avis, so results may differ with other agencies.
I was staying in central Rome, and wanted to rent a car for a week to go to the countryside. The obvious thing to do is to rent from Termini, since it is central. However, many reviews say that the line at the Termini office can often take 30-60 minutes, which sounded bad.
Instead of picking the car up at Termini, I got it instead at Ciampino airport. That's 30 minutes from the center of town, but it would have been at least a 15 minute taxi ride to Termini from where I was staying, which adds to the Termini waiting time. Taxi to Ciampino is a fixed € 31 fare from central Rome, but the rental was way cheaper (hundreds of euro), so I came out ahead there too. When we got to the Ciampino office (there is an annoying airport shuttle to get there from the taxi dropoff which maybe adds 10 minutes to the time), there was no line, and the staff was super friendly.
All in all, I feel like I came out ahead in time and money, and I didn't have to deal with the overstressed staff at Termini.