I can't imagine that its resemblance to
Delia is a coincidence. In the eighteenth century the members of Irish society who wanted to identify with the English (they were colonised at the time) found typically Irish given names an embarrassment, so they changed them to sound more posh, Latinate or mainstream. Since
Delia was fashionable, it would make sense to take
Biddy and graft it on. And for men,
Patrick became
Peter!
Alexander Pope is cheeky about one Lord
Peter or
Petre (who broke the laws of man, and God, and metre!), and he had apparently started off as an Irishman named
Patrick.
This is at best educated guesswork; but it seems convincing.