As a result of comments on my earlier question, I am posting another question.
Is there a rule of thumb for what percentage of waiting-list students is eventually placed, or invited, for a PhD position.
No, there really isn't a rule of thumb for this.
For graduate departments that admit on a fixed cycle, they generally have a certain number of places open each year. They admit from their applicant pool a number of applicants that, when all decisions are made, are designed to give them their target number of enrollees. If, for some reason, the number of students who accept is significantly smaller than the target, and there were good students who were wait-listed, they may be offered admission, although this is by no means guaranteed.
For graduate departments that do rolling admissions this is obviously not an issue, as they don't need a wait list.