She just didn't get it and more and more new accounts began showing up.
When that didn't work, I tracked down one of her relatives on Facebook, who happened to be younger so she got the whole Internet thing, and explained that her elderly grandmother (it turns out) was using an address that didn't belong to her. Her granddaughter told me she would talk to her grandma and tell her how silly she was being and promised to explain how her grandma could keep herself safe while shopping online.
No new accounts in the grandma's name have shown up since...