Minors don't require ID. Not by TSA, and not by your airline.
If your child is under 2 and will be a "lap baby," the airline will probably require proof of age so they know you're telling the truth.
Also, if my child was a mature-looking teen, I'd have some sort of ID handy so TSA doesn't think the child is 18 or over and thus in-need of ID.
A photocopy of a birth certificate should suffice (because remember, we're not talking actual legal ID), and so should an immunization record.
If your kid isn't flying as a lap baby and doesn't possibly look 18, then you'll have no need for ID, no need whatsoever.