Both the Pinocchio park and the Stibbert museum are good options, one is closer to Lucca, the other in Florence. The Stibbert is pretty cool with its knights, as Craig has pointed out.
Where are you based while at this campsite? There are also sculpture gardens, one near Pistoia, another near Siena, another south of Grosseto which I think are fun for both kids and parents alike so which one to head to depends on where you are. Other options of course include water parks like Aqua Village and amusement parks like Cavallino Matto, maybe not too cultural for the parents but fun nonetheless. These are small parks, nothing like grand amusement parks in the U.S. so very doable and not too exhausting.
Like zoos? the one in Poppi is interesting because it is all local and European animals.