I didn't read the whole thread but here is one that is a little different. It's definitely Jewish, and definitely non-Jewish origin ... from a couple of centuries ago.
A famous niggun of the Kalover Rebbe is based on an old Hungarian folk song - Szol a kakas mar (loosely translated as "the bird already sings"). When we were kids, the word "kakas" used to set us off in giggling fits for some reason.
The Kalover Rebbes's version is actually partly still with the original Hungarian lyrics. It is a beautiful niggun about longing for the Moshiach.
Kalover Rebbe here:
https://youtu.be/Us-oB6vUWWgAnd Cantor Moshe Stern here:
https://youtu.be/LHipp-A5SdQ