I think that Jim MacLean above is on more of the right track regarding the Scandinavian connection. Don't know the song but, in the context of the song, can "hinny-o" be substituted with "lassie-o"?

In the borders the use of the word "hen" for a woman, or girl, is very common, that derives from "henne" in Norwegian, Also in the North-East of Scotland, the word "Quine" is used, that derives from the Norwegian "kvinne". Could the term "hinny" be a corruption of both "henne" and "kvinne"?