Andrew Kuntz has this to say in his Fiddler's Companion:
DARK LOUGH NA GAR (Loc Dorca Na Gar). AKA - "Dark Loch na gCaor," "Dark Lochnagar." Irish, Air (6/8 time). D Major. Standard. AABB. Gearoid O' hAllmhurain believes this tune may have been learned in County Clare from Scottish sappers in the 1830's who were sent to the region as part of the British survey of the country. Whatever its origins, it became the melody of a popular 19th century song circulated on ballad sheets (the ballad gives reference to the Scottish battle of Culloden, in 1746). Recorded by Clare piper Robbie Hannon. Piper Jimmy O'Brien-Moran says it was a favorite of Willie Clancy's. O'Neill (1850), 1903/1979; No. 477, pg. 83. Piping Pig Productions PPPCD001, Jimmy O'Brien-Moran - "Seán Reid's Favourite" (1996).
P.W. Joyce also printed a version in his Old Irish Folk Music and Songs. Presumably the song referred to above is Byron's (it was widely circulated on broadsides; the Bodleian has a couple of copies), but whether or not the tune took its name from the song or vice versa, I wouldn't know. I'm inclined to suspect the former.
So far as Loch Lomond is concerned, the earliest appearance of the melody seems to have been in McGibbon's A Collection of Scots Tunes [I], 1742, where it is called Robin Cushee or Kind Robin; a little older, I should think, than Red is the Rose! (Information from Bruce Olson's website )