Hey! I think I know the answer to this! Or at least, a potential answer.

According to what I've read here on FT, airlines will sometimes issue "gate passes", which will act in the stead of a boarding pass for non-passengers to enter the sterile area. Most commonly these are issued to the parents of minors who are flying unaccompanied, and to the companions of people with special needs, to escort them to the gate or meet them at arrival.

Call the airline that your son is travelling with, explain the circumstances, and request gate passes. My guess is that they will limit the number - you might only get two - but it will allow you to enter the sterile area and wait for your son's flight, grab a quick bite, and walk with him to his connecting flight.