well it appears based on what others are reporting that if there aren't any days left on the ticket, you're going to be offered a one day ticket as your only option. And according to the rules, that ticket MUST be used that day.
SOOOOOOOOOO, if I were your niece, before heading off to GS at one of the parks, I would just add a day to the child's ticket at the front desk of your hotel. If they already have multi-day tickets, it only going to be a couple of dollars. That way when they scan the ticket, it will still show an unused day, and he would be able to get the Bday fun card. Even though the ticket will not get used it's basically like buying the Bday fun card (valued at $68 for a child per an earlier poster) for like $3 or $4 for the extra day. Not a bad return on your investment IMO.

Hope they have a wonderful trip!