I've always interpreted "bisschen" as just meaning "a little".
Does it come from the diminuative of "der Biss" (the bite)?
The German Wiktionary writes about its linguistic origin with reference to Johann Christoph Adelung: Grammatisch-kritisches Wörterbuch der hochdeutschen Mundart:
entstanden aus der Diminutivform des Substantivs Biss; es bedeutet somit also: Gerade soviel, wie man auf einmal (mit einem Biss) abbeißt. Adelung stellt den Wandel der Diminutivform „Bißchen“ (Ein Bißchen Brot / Ein Bißchen Wein) zum Nebenwort „bischen“ dar
So, it comes from the diminutive form of „der Biss“: das Bisschen = little bite
Actually both meaning, and usage of "ein bisschen" is very similar to the English "a bit" (vs. "bit/bite"). Likewise we also know the "little bit" in German: "ein kleines bisschen". Note that it originated form a diminutive of "Biss" but it is not used as a noun in this context, seen by the lowercase spelling in its present usage as a pronoun. When used as a noun "ein Bisschen" in the meaning of a little bite we have to spell it with an uppercase initial.
Both the English bit, and the German Biss share the same etymologic root with the Indo-European bheid- (splitting something with an axe). Interestingly in Swiss German "Mundart" the meaning of Biss was a wedge.
Regionally there is quite some variation for bisschen:
bissel
bisserl
bissle
chli biss
...and many more
Perhaps worth noting:
The Dutch language, which is very close to German, has the exact same construction with the same meaning and etymology:
beet = bite. een beetje = a little bite.