We have a 3-YO son and based upon our experiences, we have been able to get 98% of the items through security.
Length of Trip: If the flight is only one hour long, don't bring the whole refrigerator with you...they are not going to allow it. We were going to LAS and my wife just brought too much food\liquid items and the TSA disallow most of the items.
If your flight is only an hour but you connect to a 10-hr flight...show the TSA agent your boarding pass/etc.
A few times, the TSA agent asked about the quantity and we said that we are at the airport 2 hours before our flight and our flight is 4 hours.
Disclosure\Organize: We never have any problem when you put all of the liquid items, ice paks and into a single tray. Once my wife left a bottle in the bag and it was confiscated.
Attitude: Be nice.
2nd Opinion\Ask for the Supervisor: A few times, we have asked for the supervisor when the TSA agent wanted to confiscate our items. Every time, the supervisors allowed the items through security.