This past weekend we took a family date to the Bargello. The big open courtyard on the first floor is a great spot to let Livia run around and explore. She would occasionally stop and stare up at the sculptures in awe (some of them are massive) and then turn around and run in the other direction. We got just enough energy out in the courtyard that she was quiet as we explored some of the other rooms. She *loved* the pottery room and was mesmerized as she looked through the glass at everything -- although all I could think was, "I feel so bad for the person that needs to clean the glass tonight." Fingerprints galore!
In my opinion, the Bargello is a great museum to visit with kids because there are a lot of spaces for them to roam, catch a breath and reset. It's also typically not very crowded so you don't have to worry about your kid getting too overwhelmed by crowds or, even worse, lost in the crowd. Plus, I think sculpture more so than painting is a bit more interesting to really young kids because it seems a bit more lifelike.