Cortona?

Cortona would be quite a detour. From Cinque Terre to Rome you should stick to the coastal rail route.
Lucca is not exactly along the route, but a short detour (from Viareggio). Further south you could consider Porto Santo Stefano (not exactly on the rail line but a very short bus link away), on the Mt. Argentario peninsula and from there even take a day trip to the island of Giglio.

ln,

The fact that Cortona is not listed in the RS Italy 2016 guidebook may be an indication that it's probably not the best place to spend two nights. It's a very scenic small town with lots of interesting shops and restaurants, but doesn't provide a lot of activities. There's the requisite Duomo and a few small Museums, but not at lot other than that.

As Roberto mentioned, Cortona is also somewhat out-of-the-way on the route from the Cinque Terre to Rome. If you pass inland from La Spezia, you could perhaps spend two nights in Siena or continue a bit further and spend two nights in Orvieto. There are lots of possibilities.

You could also consider staying in Pisa (along the rail line to Rome, and visit Lucca from there (20 min from Pisa by train or bus). That way you don't have to carry luggage on and off trains too much. Just check in a hotel in Pisa near the station, visit Pisa in the afternoon,, then the next day do your day trip to Lucca from there without being burdened by luggage. If you travel in summer Pisa is also close to beaches at Marina Di Pisa and Quercianella or Castiglioncello (both very nice).

I have stayed in Cortona and enjoyed the town,we day tripped the region from there,also stayed in Orvieto and liked it,beautiful cathedral and museum,both of those towns are worth staying and I would go back.Lucca is a great place to visit,plenty to do,I thought that Pisa has one main tourist area the Field of Miracles and it has the tacky vendors across from the Leaning Tower.
Mike

I would opt for Mount Argentario. You may stop at Orbetello Scalo with the train and take bus to Porto Ercole and Porto Dan Stefano. The whole area is fantastic if you love nature and birds.
A trip to isle of Giglio is also wonderful.

Based on your route and my enjoyment of each destination, I recommend Volterra. If you are interested in visiting an Etruscan museum, the one in Volterra is much more comprehensive than Cortona's. Cortona is only in Rick's Florence/Tuscany book, while Volterra and Lucca are in both books and visited on our Heart of Italy tour. Lucca is flat, not a hilltown, which makes for easier walking but less view.

Volterra is wonderful!