In October my wife & I had the opportunity to spend 6 wonderful days in Florence, staying at a great AirBnB, 15 minutes from the Academia.
On our first evening, it was late and we found the Il Micio by accident, which was only 2 blocks from our apt. It was a wonderful surprise, only about 8 tables and from what I could tell all locals, including our landlord from our AirBnb.
It is a family owned and run restaurant, which was both comfortable, great food and very affordable. In Florence you have a choice of many restaurants, but we ended up going back to Il Micio two more times, each time a great experience.
Would highly recommend to anyone looking for great food, good prices and nice family / neighborhood feeling.