The story
The pronoun me in this expression is not semantically vacuous beyond its reference to the first person singular (and thus simply equated with the subject by copular be, as in this is me), but a relational dative (literally woe is to me, i.e. woe is upon me; compare German weh mir). Cognate with Yiddish וויי איז מיר (vey iz mir).
Etymology from Wiktionary, CC BY-SA 4.0.