Always wondered if somebody traced the history of how an old Talmudic phrase was put to Lead Belly’s upbeat tune, and said tune travelled around Purim celebrations to now become a pervasive old-time Purim favorite.
Here are some Orthodox Jews singing the Purim version.
Here’s Lead Belly’s original.
Man, the world gets bigger every day, don’t it? (I don’t have an answer for you, I’m just amazed that “Pick a Bale of Cotton” is evidently a Jewish song.)
No answer, but that’s news to me! Who knew?
For fun, check out this. It’s not just Mishenichnas Adar.
I’m guessing that the Finding Nemo hat worn by one of the singers isn’t terribly traditional either. 
Leadbelly was one of the first big name draws in folk music, back in the 30s and 40s when the audience was mostly limited to New York and heavily Jewish.
I’ve corresponded with that blog’s author in the past (we were both part of the ex-Orthodox-Jew-o-sphere 
). He certainly put together quite a collection there!
Back in Eastern Europe, Jews would take folk songs or drinking songs and just slip Jewish lyrics in; it wasn’t so abnormal, and we could probably find a bunch of examples of that, but I don’t think it happened much in America. A lot of the songs referred to at the 2nd Son blog are relatively new. Modern secular songs have been ripped off by some Orthodox band/singer and because the Orthodox band was popular, the songs get sung regularly.
“Pick a Bale of Cotton” is interesting to me because it’s around in America in the first half of the 20th century, so the origin of the Jewish version is more mysterious…Jews aren’t taking secular drinking songs like they used to, and there aren’t exactly a lot of Orthodox bands. Was Lead Belly’s tune originally picked up by a cantor, and did it just spread like wildfire? Did some young peoples’ group decide to introduce folk into services? I’m intrigued.
So what are they saying instead of the original lyrics?
So you are saying that “Pick a Bale of Cotton” is Talmudic? Was there a phase of Jewish history when a lot of Jews were employed as cotton pickers? 
Haven’t you ever heard of the old Jewish spiritual “Go Down Moses”?
Not to be confused with “Go Down On Moses”, which just ain’t kosher.
It’s Purim!
The little girl has a princess dress on.
It’s dress up in costume like Halloween.
I wonder if it has to do with Jewish involvement in the Civil Rights movement.
True dat. Alan Arkin’s father was a big part of this (in the 50s, mainly), and the connection is well parodied in A Mighty Wind, e.g. when a very non-Jewish producer tries to ingratiate the musicians by sprinkling his speech with Yiddish words.
Great, now I’ve got Allan Sherman running through my head:
(For those too young to remember, Allan Sherman was a song parodist from the '60s whose schtick was to apply Jewish references to more-or-less traditional songs. So this was Leadbelly set in the Garment District.)
They’re just repeating the Talmudic phrase, “Mishenichnas Adar Marbeh BeSimcha,”
which means “When the month of Adar comes, we increase our joy.”
I don’t know about the Hassidic version, but the Israeli army has its own lyrics (translated):
Combat engineering are waking through a minefield
Combat engineering are walking all day long
Oh-oh-oh there goes my leg
Oh-oh-oh there goes my arm
No, no, no… you want a Jewish work song, it’s:
"Gonna see Miss Liza… gonna go to Mississippi…
“Gonna see Miss Liza… gonna go to Mississippi…”