What do you think the teacher's brother will do with Don's business card? To me, giving the card to the young guy looks like an invitation to blackmail. But from experience I'm guessing the Matthew Weiner is not going the "obvious" route.

Any thoughts?

It's definitely in the story for a reason.  Maybe the brother will present himself at Sterling Cooper for a job.

But what can he do at SC? He is adamant that he's not going to spend the rest of his life "cleaning toilets" i.e. doing menial work.