For the record...
Having grew up just 70 miles east of NO, I have some knowledge regarding this topic. In the late 50's I can remember we use to often times leave the front door open with just a screen door between us and outside. When someone would knock on our door, it was not uncommon to hear,
Me: "Who dat?"
Response: "Who dat?"
Me: "WHO DAT SAY WHO DAT WHEN I SAY WHO DAT!!"
To me...this is where it all started. Everybody who grew up down there knew this little ditty.
Also...it is part of the dialect of that area...dropping the "th" in the front of a word and making it a "d". I can remember well my mother correcting me while smiling when I would say "dis", "dat", "dem", "dey", etc.