The TSA's page on Identification for flights states:
Adult passengers (18 and over) are required to show a U.S. federal or
state-issued photo ID in order to be allowed to go through the
checkpoint and onto their flight.
So that would imply that minors don't have to have ID.
However, to further confirm this, I'll quote from an airline - JetBlue's page:
Children under the age of 18 and traveling on a domestic flight do not
need to show identification or documentation unless they are under 14
days old (physician’s letter required) or are traveling as a lap child
(proof of age may be required).
and since you didn't specify if they were travelling with an adult:
Unaccompanied minors must check in at the ticket counter. An
unaccompanied minor does not need photo ID. However, the adult
dropping them off and picking them up will need photo ID. This adult
must be named in the reservation.