You must definitely stay at Eleonora’s apartment. It is in a beautiful neighborhood away from the noise of Centro Florence. If you only have a little luggage I definitely recommend walking from the train station to her house. It is about half an hour walk but you get to experience the city for the first time and it is free.
When you arrive at her place it is quiet and very beautiful. The apartment is very thoughtfully decorated and enjoyable and the fridge is filled with goodies.
And just down the hill from her place there are some delicious local restaurants where you do not find many tourists.