According to this 1941 Life Magazine issue, teenage girls in Atlanta were making up rhyming pairs like this at the time under the name "stinky pinky". https://archive.org/details/Life-1941-01-27-Vol-10-No-4/mode... Webster's Dictionary from the 60s has the game listed under that name, https://archive.org/details/webstersthirdnew0000phil_l0b1/mo... and that name also seems to continue to today, e.g. by the radio show Loveline.
It's possible I found this decades ago and the origin of how I learned this game was lost to time :)
i have five clean references to "as a mason jar" so my collection is fairly complete ;-)
note: ripgrep 4.079s wall; ag (silversearcher) 5.916s wall; grep 6.940s wall
Sorry about that.
https://boardgamegeek.com/boardgame/8785/brain-strain
I made a proof-of-concept daily game: https://awfulwaffle.jonabrams.com/