Florence to 5Terre or Portofino

Pisa has luggage lockers. But La Spezia is a known place where you could drop a car off and then take the train to portofino or a town in Cinque Terre. Just use Google maps to see if the same car rental companies have offices in the towns where you what to leave from and go to.

I'd suggest staying in one of the five towns of the Cinque Terre, rather than Portofino. If you're not able to find a vacancy, you could also try Levanto, which is only five minutes or so by train from Monterosso. Portofino is fine to visit for a few hours, but there's not a lot to see there. Staying in the C.T. will also make your ongoing trip easier, as Portofino doesn't have a rail station (you'd have to use Santa Margherita Ligure, and then take the boat to Portofino).

I would not suggest renting a car for the trip you've outlined. While you could return it in La Spezia, there are a number of potentially expensive caveats to be aware of. Each driver listed on the rental form will require the compulsory International Driver's Permit, which is used in conjunction with your home D.L. You'd also have to be extremely vigilant to avoid the ZTL (limited traffic) areas that exist in many Italian towns. Each pass through one will result in hefty fines, which you may not know about until after you return home. Florence is just about saturated with automated ZTL cameras, so not a good idea to drive there.

The hassle of renting a car and driving in this area should be worse than the hassle of stopping off the train at Pisa, which has luggage storage. Disclaimer: I have never driven in this region, but I would not want to, personally, having to worry about ZTL zones etc to avoid getting fined for driving in the wrong part of town. I know you can drop a car at La Spezia or Genoa, but the train is pretty easy. You would have to take a ferry or bus (or taxi?) from the Santa Margherita Ligure train station, however, that still sounds less of a hassle to me than driving.

I have driven in this area, when I already had the car for more rural destinations, and also recommend against renting a car for this plan. ZTLs were not an issue, but traffic, one-way streets, having to pay for parking, and probably having to drop the car a fair distance from the La Spezia train station all can be hassle factors.

Rather than staying in Portofino, if you decide against staying in the Cinque Terre, I recommend Santa Margherita Ligure, a beautiful small city with more to offer than Portofino. Each time we have to Portofino, we spent two hours there at most.