I am sorry to be the one to tell you that Florence, unfortunately, does not offer a museum pass that would make it easy for your family to visit several museums and thus save a bit.
It gets worse from an economic standpoint - since it seems you'll be in Florence on June 2, a new temporary exhibition that starts on May 30 at the Uffizi will make the base price go up to 10 euros - if you were thinking of booking ahead, the total cost would be 14 euros per person (4 euros booking charge per ticket). More info on the special exhibition here: http://www.polomuseale.firenze.it/english/musei/uffizi/
The closest thing that comes to a "pass" is to become a "Friend of the Uffizi" - a family membership is only for 2 adults and two minors and costs 100 euro. If you were staying in Florence for a few days and visiting several museums, it could pay off (theoretically, if you were planning on visiting several museums). With this membership, you get to visit all of the state museums for a whole year (I know, you wouldn't exploit it beyond your short visit regardless). You can view the whole list of museums and read more here: http://www.amicidegliuffizi.it/become_a_member.aspx
However, as it seems you were just planning on spending one day in Florence, then this would definitely not work for your family so I am afraid you'll end up paying the full price per family member.