I stayed at the Hotel Sampaoli when I was in Firenze two years ago, and I enjoyed immensely. It's in an old building about fifteen minutes from the train station (you can get a bus to Piazza San Marco if you don't want to walk), and it's fairly close to all the museums. Rooms run the gamut from singles to an all girls dormitory, and they are all clean and nicely furnished. The major perks are that you get keys to the front door, so you don't have to worry about a curfew, and they provide a computer with free wireless access.

I stayed in the dormitory, and I really loved meeting people from all over the world. I also loved that it had a bathroom and a shower of its own, which was completely unexpected and very welcome!

The only gripes I think people might have with this hostel are (a) that you have to drag your luggage up three flights of stairs to get to the rooms, and (b) they insist that you pay your bill in cash. However, the rates are extremely reasonable, and they might discount your bill if you stay longer than a week, which is what they did for me. And Florence is largely a cash city, unless things have changed since I've been there, so it's not that unusual.

This is the Sampaoli's website, but they also offer very good rates on Hostel World.