Liland, there really is no way I can see this as related to "adonay". First of all, despite its English translation, that word is not used as we might say, "My Lord", in the sense of crying out to God when we're in distress. It's only used in the synagogue liturgy and in formal blessings. There are many other phrases that are used when "informally" addressing God, and which would have been used by an Eastern European poet if that were the context.

In addition, the transliteration given as "donaj" (which might understandably lead to that derivation) is simply incorrect. The word in the Yiddish text would be transliterated as "dawnah" or "doonah", depending on your accent of choice. The final vowel is "patach alef", which is pronounced "ah", not "ay", "ai", or "aj". In addition, Yiddish words that are originally Hebrew use the Hebrew spelling (which often is incongruous in Yiddish), and this is definitely a Yiddish spelling. It's not a common form for a Yiddish word, though, and looks more like a transliteration itself.

Which raises the interesting possibility that it may have come from a different language...which might make ddw's original guess not quite so out of line, after all! I'd be interested to know if anyone has ever pursued this further...did the theatrical song come from a folksong in yet another tradition? Very interesting...

Aloha,
Mark