Hi, registered just to reply to this. I've just taken my mum, Martha (who everyone calls Pat) to place a wreath on my grandma's grave. My grandma's also called Martha. Her headstone actually reads "Martha (Pat) Hatton"
I've always just taken the fact that everyone, including my grandad, calls my mum Pat. Never questioned why. Googled it and it brought me to this forum.
So I guess I'd say that if there was an interchangeable name for Martha then Pat is definitely one.