The name means nothing since the Samuel Lover and also the Newfoundland songs both have "Molly Bawn.' I did look at Bunting, the source of my comment however is Andrew Kuntz, not corroborated by me, although I know Andrew and he's pretty accurate. I'm talking about the melody, Lough Sheeling about which Patrick Joyce said, "My version is just as I learned it from the intelligent singers of my early days. The air is the same as "Lough Sheeling" of Moore's song, "Come, rest on this bosom!" Joyce's version is nearly Identical to the 1797 Irish broadside- at least for two stanzas!!! I'll post later.
Brian, whether it's of Irish origin is not known, probably just assumed since it was popular in Ireland (according to Joyce born there in 1827) and the names and places point to Ireland. What makes you think it could be English?
According to Andrew Kuntz: "The title 'Molly Bawn' is an Englished corruption of the Gaelic 'Mailí Bhán,' or Fair Mary (Fairhaired Mary, White Haired Mary)."
That might lend some credence to an English origin.