How did the sweatdrop originate? I can kind of handwave or fanwank the origin of things like the exasperated faceplant or the vein popping out when a character is comedic-angry, and even the trope of Japanese flag fans coming out when people want to cheer on their comrades, but how did the sweatdrop come to characterize the feeling of sort of embarrassed contrition? 



I can't come up with anything logical just based on speculation. Is this an outgrowth of the manzai tradition, with the tsukkomi and boke characters?
There doesn't seem to be any "official" announcement for that, but I think it's safe to say that since Manga and Anime make heavy use of symbols to re-create a concept or a situation, the sweatdrop functions the same way. In other words sweatdrop (see image below) is a reference to the fact that when people feel uneasy or feel similar emotions, they usually get sweaty. You don't get that sweaty in real life, but again, they're symbols.  

More symbols that might prove this point:


The cruciform popping veins are a clear reference to the veins that pop out and can be seen pulsating when you're nervous or you're about to lose it.


 Source


The red cheeks when you're embarrassed (blood flowing), the red nose when you're drunk (yes, nose has that tendency sometimes when you get drunk).
The saliva drop when you're hungry. Your mouth does get watery in that situation, because it anticipates the food you're about to eat. Mastication is the first stage of digestion.
The eyes shaped like >.&lt; that indicate nervousness or excitement. (I bet everybody can picture this face expression).


And so on. There is more (as a start) on the Wikipedia page for Manga Iconography.