"Remind me to make a doctors appointment late next September."
She almost got it right. This resulted in a reminder to make a doctors appointment, but the late part apparently didn't get handled, as it was set for midnight on September 1st. So a switch over to the reminder (fortunately, the one she created was clickable right from the Siri window, so I didn't have to back out of Siri, swipe over to the Reminders app, and dig back through to find it), and change the date to later in the month.
I was wondering if she would somehow select some date in the back half of the month for late, but I guess not. What about the end of next September? Maybe that would have worked better?