We've sent our kids with their aunts/uncles (different last name) on several occasions, and never had an issue. You can't even book the minor reservation without tying it to some sort of adult reservation typically -- unless you are specifically booking/paying for an unaccompanied minor reservation. So it's just assumed that whatever adult reservation they've been booked on has consent to travel with the child. And that person is necessarily identifiable by virtue of having a ticket.

The major exception to this would be international travel -- in which case the child needs to be traveling with their parent, if not both parents. And this is primarily to avoid parental abductions -- i.e. one party in a divorce fleeing to Mexico with their child to escape local jurisdictions.

The one thing I would recommend is that your friend call up Southwest and see if she can get her son's minor reservation re-associated/linked with your adult reservation. Otherwise they may not let her son check in without her present.