In the magazine business, blow-in cards are the annoying loose cards that fall out of every magazine that you subscribe to. (Yes, they have a name. I was surprised too. Everything has a name.) Curiously, most magazines use these as opportunities to send you five subscription cards to a magazine you're already getting, just in case your kids, significant other or dog need to subscribe as well, and don't feel comfortable phoning it in. The name comes (I believe) from the process by which the cards are inserted into magazines prior to distribution.
Compare bind-in card.