My wife and I will be in Florence in early October. We will arrive by train on Sunday the 8th and leave for Rome early Friday the 13th. We expect to tour Florence Sunday and Monday, then have tickets on a transportation provided tour of Cinque Terre (boat included) on Tuesday. Wednesday and Thursday we would like to explore Tuscany by rental car. Perhaps one day to Pisa/Lucca area and the second day to Sienna/San Gimignano. It has been suggested that we could get a hotel for the 5 nights near a train station relatively close to Florence at which we can park the rental car, and use the train to go in an out of Florence. Does that make sense? Suggestions?
Sure. But if it were me, I would visit Pisa/Lucca by train because it is just so easy, and then book a private driver for Siena/SG--maybe the place you booked for the CT tour has one on offer. I love staying in small towns in Italy with a car, but in this case, I think I'd opt for enjoying Florence more intimately (then maybe next time, you'd explore rural Tuscany).
A lot of back tracking in that plan, since you will pass Pisa and Lucca on your trip to Cinque Terre. And overall a lot of ground to cover in just three days.
If I were going to see the same towns by car, I would drive to Pisa or Lucca the first day and stay in a hotel that provided parking, visiting the other town on the way. For CT, drive and park at La Spezia, then train or boat to CT.
Then drive toward Siena. Or, reverse direction and start with Siena/San Gimignano. Then arrange to drop the car at one of the western towns, and take train to Rome from there.
Of course, the packaged tour to CT would need to be refundable to make an alternate plan work.
There is actually a tour company called Walk About Florence that provides an all day tour of Sienna, San Gimignano and Pisa and lunch at an organic winery. It is just a taste of Tuscany but a really nice day and a great value. It will probably be similar to your Cinque Terre day trip, as that will give you a quick overview of that area and you for sure will want to go back to stay and really visit the area.
No need for a rental car.
Pisa+Lucca are best done by train (go to Lucca first, which is prettier, than end the day n Pisa) so you don't have to deal with traffic and parking. So basically you do a loop: Florence to Lucca (90 min) - Lucca to Pisa (30 min) - Pisa to Florence (50 to 80 min depending on train). Trains start early in the morning and end late at night. There is at least 3 trains every hour between those cities.
Siena can be visited by bus or train from Florence (the bus is 15 min faster and the bus station is on top of the hill in Siena downtown, while the Siena train station requires a taxi to downtown on top of the hill).
San Gimignano is more or less between Florence and Siena (with a slight detour). There are buses from Siena to San Gimignano and San Gimignano to Florence (bus change required at Poggibonsi-SG station). Doing both Siena and San Gimignano on the same day requires an early start. Go to Siena first, then if you have time you can return by detouring via SG. You can visit SG in a couple of hours max.
You can also use an organized tour with a private driver (maybe a small group or just you and your spouse). That would be the most efficient option, albeit more expensive than the bus option. Renting a car would be slightly cheaper than the private driver, but once you add the logistics of picking up the car, driving, getting lost on the way, finding parking, returning the car, it will probably take you just as long as going by bus.