A few places that sprung to mind, just to seed the thread -
Osteria Cipolla Rossa
Sergio Gozzi (only open for lunch)
Enoteca Pitti Gola e Cantina
Baldovino (I think this place feels like it should be a tourist trap, and somehow isn't)
All'Antico Vinaio (line's gonna be long at regular meal times)
La Casalinga (stick to the daily menu - the items in CAPS)
Perche No
Carapina