Remove Faces Permanently from Photos Library
The iOS 12 and macOS 10.14 versions of Photos makes this significantly easier.

Select the face / person / name in question and then select:


Remove this person (from macOS)
Remove (from iOS)


That purges the name and face from ones it will select in the People Album. I do see that both platforms will find names from your contacts, so if you just removed "Mary" from your album and you have any contacts with Mary in them, you will still get prompted. Temporarily deleting the contacts (make a backup) or deleting the sync of cloud contacts like iCloud and Outlook.com (or the many other contact sync options) you can test that Photos doesn't have that name anymore to suggest to you.