Here's a clue
I have two birthday calendars in the Other category -- same symptoms. But, some time ago I modified the name of one of my contacts to remove his middle name. Now, this birthday appears once with his middle name and once without his middle name. I cannot find an entry anywhere in my iPad with his middle name (except in the extraneous birthday calendar).
So here's my guess: somehow during an iCloud merge / sync the old birthday calendar (the one with a middle name) became disassociated with my Contacts. It is automatically backed up every time I sync and it is restored if I reset my iPad and sync from iCloud backup. But because it is unrelated to Contacts, I cannot deleted the obnoxious birthday calendar.
Does anyone know how to perform a calendarectomy on iPad backups?