I always like to know WHEN and WHERE something originated before answering how.
That said, I asked this very same question in the late 90's and was told that it came from older video games with characters that were so small in size that they couldn't do any facial expressions with them.
It sounded reasonable to me, because I can't think of any animes in the 80's using this trope, but doing a little bit of backtracking I can't find any early video games using sweat drops either.
I do know the September 1995 Sailormoon RPG has it, but that's too now compared to manga and anime examples.
The oldest anime example I now know of is (yet again) Sailormoon, right off the bat from the first episode, March 1992.
The oldest manga example is Sailor V, right from the start again, thus making this trope start back at at least August 1991.
However, the vein popping trope, also present in Sailor V 2nd volume is older, being at least as old as Super Mario World which was released in November 1990.
So there might be even older examples than this. If anyone can state older examples, please do so.