You have just half a day, so your opportunities are limited. I do recommend the activities that the Palazzo Vecchio organise for children, as Eugene has mentioned above, the Secret Passageways tour always seems to go down well with kids. This does need to be pre-booked.
My kids always enjoyed climbing the dome of the Cathedral. The queues to this can get quite long, so it is a good idea to do this early in the day. They also always want to go and see David at the Accademia.
There is the small Museo di Leonardo da Vinci, near the Duomo, which has replicas of over 50 of Leonardo's wooden machines. Most can be played with to see how they worked. My kids love shopping on San Lorenzo market and the Palazzo Davenzati is another favourite - a restoration of a Florentine merchant's house, a colourful maze of brightly decorated rooms.
If your child is into science there is the Museo Galileo - full of historical scientific instruments. If your child is into clothes, there is the Costume Gallery in the Pitti Palace.
I should also mention the great programs they run at the Palazzo Strozzi to get kids involved with the art exhibitions they run there and the Natural History Museum, La Specola.
I hope there is something in there that will appeal.