Your son doesn't happen to have a white focus st that hes putting blue vinyl all over does he?

Did they already give him the title for the new car?  Did he title it with the state?  If it's been 30 days I assume he did.

Dealership could probably sue your son as he didn't give them a clear & signed title.  Shame on the dealership for not making sure both parties signed; however, just like in real estate, I'm SURE the contract he signed has some mention that if there is a clerical error (missing signature, wrong date, etc.) that both signors promise to mutually work to correct the error.  This isn't going to be an "oops, they got us" deal.....they'll do what they have to get that signature.  No doubt the contract said something about how he certifies their are no outstanding leans, loans, etc and he passes CLEAR title to them......without that signature it's kind of a lie (unless everybody can say oops and mom signs it.)

If he's had the new car that long, he was approved for the credit and that part is done. There is no reason for your wife not to sign over the trade-in.

They are gonna void the deal and come get the car from him if she doesn't go up there and sign the title.

Don't sign it and fuck him over now, or sign it and leave him to his own adulthood.

And, if they void the deal, they will report it to the credit agencies and screw his credit over.  They could also file a small claims suit for the devaluation of their new value.  It would be in the kids best interest for the wife to sign over the car and leave him to his fate.

Doesn’t sound like a good outcome either way.
Not sure if he still lives at home, but the way I was raised; if you want to make big boy decisions like this, you’re old enough to move out. Start packin your shit.