Visiting our daughter studying in Florence in February - thoughts on where to go

I'd consider larger towns like Siena and Lucca, but not necessarily plan on driving around to villages. It sounds like you don't necessarily need to stick close to Florence, so I'd also consider Bologna, Torino, Genoa.

I would suggest Siena and Lucca - both would be good to visit any time of year, and both are easy to reach by public transportation (train for Lucca, train or, preferably, bus for Siena).

I remember our landlady on Lake Como talking about fog there in the winter but I have not been there then to confirm.
I recommend that you visit Bologna, a university town with a food focus. You could do day trips to Parma, Modena and Ferrara.

Assisi is a bit of a distance away but definitely worth a visit. It took us about 3 hours by train from Florence, but I loved it there and thought it was so worth the trip. The town is incredibly charming and those views of the Umbrian countryside were breathtaking.

Lake como towns will be basically shut down in Feb, ferries will be running but I would skip them

Definitely go to Siena
And any of the other larger hill towns like Lucca, San Gimignano will be good to visit at this time

You definitely want to visit Siena by direct bus (1h 15m) that will drop you off in the old town. The train takes longer and then you have to take a bus from the train station to the old town.
From Florence’s SM Novella station take a direct train to Lucca but there aren’t many so get an early start (1h 30m) https://www.trenitalia.com/en.html.
Pisa is another good day trip from Florence (1h) and other’s have pointed out so is Bologna but it’s important you know what you want to experience because Bologna is a big city so have a plan before going (45m). Parma and Modena require a train transfer (2h for Parma and 2h 15m for Modena). Ferrara does have a direct train but there aren’t many and also requires an early start (1h 15m).
A direct train to Torino takes 3h 15m so this is not a good day trip option and neither is Genoa that requires a connection that also takes 3h 15m.
Assisi takes 3h and requires a connection and is better as a day trip from Rome than Florence. Skip lake Como that also requires a transfer and takes too long to get to and back in a day.
Once a I took a day trip from Florence to San Gimignano that requires a connection by bus and it was a very long day. You need a car.

If you like cathedrals, Orvieto.

If you like mosaics, Ravenna.

The Siena Cathedral is spectacular also.