I can recommend the place I just stayed. It's cheap (45 Eur/night for a single room, which turned out to have a double bed) but the location may not be ideal just because it's a long walk to/from the train station and not in an exciting area. But once I figured out there's a frequent bus and where the stops were, the location didn't matter so much.

For one night I might just splurge on something more central in Florence. But if you are interested, the place is the Bed and Bed Cassia - I just reviewed it on Trip Advisor:

http://www.bedinflorence.it

Sister property Bed and Bed Peterson might be slightly closer to the train station.