Solution 1:

I think that David M's Comments explanation ("It probably started as a way of being cute and got out of hand.") is exactly on point. If someone writes

Because donuts.

the phrase amounts to an extreme telescoping of a longer (usually jocular) idea along the lines of

Because donuts are a consideration here—and when donuts are involved, nothing more need be said. I mean, think about it: donuts, for gosh sakes. OMG let's stop talking and get some donuts! Mmmmm, donuts!

This telegraphic style—with its omission of one, several, or many surrounding words—may or may not be linked to Twitter tweeting, but I don't see any necessary connection between the two. The first time I can recall encountering a similar wording involves Patti Smith's 1978 song "Because the Night"; in the full lyrics, the relevant wording turns out to be

Because the night belongs to lovers. Because the night belongs to lust.

etc.; but the title itself was startling and provocative because it stopped at "Because the Night."

Antecedents aside, I don't think there's much point in trying to assign a part-of-speech identity to because in a "Because noun" formulation, since that formulation reduces the underlying idea to a vestigial remnant.