Two quick recommendations for places near the Boboli Gardens:
Volume (cafe) in the Santo Spirito - a really hip place, friendly staff, great music, good coffee, free wifi, clean bathrooms. All of those things make for a great experience! Highly recommended.
The restaurant immediately adjacent to Volume (has a porch). We ate outside on a warm April day; enjoyed our pizza (cliche, but it was tasty) and salad tremendously. Try the deserts! Frequented by other internationals, and by Italians.
So - two spots close to Boboli, central, but quiet (spirited away from the crowds, perhaps?).