One of my favorite spots for cheap food in Florence is called I Fratellini (also called I Due Fratellini). The two brothers that run this place (hence the name, translated to "the two brothers") whip up delicious paninis super quick and for really cheap-- most being only 2.50 euros. You can also grab a cheap cup of wine to drink while you chomp down one of the many types of sandwiches they offer.
There is ALWAYS a line around lunchtime, but worth the wait!
My favorite is #6, pomodoro e mozzarella (tomato and mozzarella), and they usually will throw some basil on there for me too!
I Fratellini is in between the Piazza Della Repubblica and Piazza Della Signoria, on Via dei Cimatori.