Wait - there is no site.
I just tried googling what I assume you did and came up with some .org thing - I'd steer clear.
Student rush tix are like lottery tix, but you often have a better shot at them because you need a student ID (the person with the student ID can buy 2 tix under that ID, the other person doesn't need a student ID).
You don't need to sign up for anything.
Some shows don't do student rush, but a lot do. Some have slightly different policies.
Most you go to the box office when it opens in the am or in the afternoon (some start the deal in the morning, some like 3 hours before curtain, official show websites or playbill.com should have specifics per show), and if there are tickets available, you can have them for like $30 a piece. They're often very, very good seats. Best bargain on B'way.
Same as lottery rush, which you do the same way and they draw names from a hat from everyone put their names in, right before the show. That, you don't need any student ID for, so it's more competitive, but you can put into one show's lottery, and your son could put into another and then you each stand one place and hope one of you wins.
Here, found the Playbill linky - that lists most shows' policies, whether they do student rush and what time to show up for student or lottery.