The word "just" originates from "justice", but when used as an adverb, it means something different (but similar) to its adjective counterpart.
When did "just" become used as an adverb in American English?
(Correct me if I'm wrong, but I don't believe "just" is typically used as an adverb in British English.)
Just as an adverb meaning barely or merely has been a part of English since the 1600s. It wasn't a new word then - it was already an adverb in Middle English, where it had the meaning of "precisely" and "punctually" since at least the 1400s. The meaning just shifted between the 1400s and 1600s.
The use of just as an adverb in Middle English in the 1400s came from the adjective form of "just", introduced into English in the 1100s from Old French.
Source for the above: The Online Etymological Dictionary.
Conclusion: "Just" as an adverb meaning "barely", "only", "recently" is not an invention of American English; it has been used with the same meaning for about 400 years, and used as an adverb with a slightly different meaning for a few hundred years before that.
I think you'll have much better chances asking this in /r/linguistics.
Also, which "just" do you mean? The one denoting closeness/similarity? (I just arrived, This is just what I was looking for)