There are no rental locations in the Cinque Terre, so that isn't an option, but La Spezia is just a few minutes away by train.
If you are going to Siena, I don't recommend to go all the way to Genoa to pick up the rental car, as suggested above, because Genoa is in the opposite direction from your intended destination and much farther away from the Cinque Terre, than La Spezia.
Regarding the toll roads, they won't impact your budget much at all. If you go to Siena and take the A12 to the A11 to the A1, the total toll will be €13.90. But you could also choose to take the FI-PI-LI Freeway (which takes slightly longer since the speed limit is lower, but is toll free) and maybe stop in Volterra or San Gimignano on the way to Siena, then you would be on a toll freeway only from La Spezia to Pisa Airport exit, and the toll would be only €8.30.
Most of the driving throughout Tuscany, is likely to be on regular roads, not toll roads, because the famous hill towns aren't really along the A1 or the A11 toll roads. Even those which are (like Lucca or even Cortona) can be reached through alternate toll-free highways.