Right: A collection of these lyrical poems is at The Internet Archive: The Irish melodies, national airs, sacred songs, etc., of Thomas Moore (1874).
On page 200 is what appears to be the original introduction, which explains the book added lyrics to pre-existing airs that either had not lyrics or lyrics that would be unintelligible to most. On page 212 is "Oft in the Stilly Night". The only provenance given is "(Scotch Air.)"

Of the 76 poems from National Airs, in only one case original words are mentioned, and then only as the inspiration for the poem. In three other cases the original titles are given. Other collections that include National Airs seem to differ little in this respect. Unless it turns out the publication that included the scores had more information, we'll have to assume that in the other cases Thomas Moore didn't know the originals, or they were unintelligible to him.
(There's an 1860 edition with music in Google Books, but it has even less information than the edition linked above.)

It would seem the melody of "Oft in the Stilly Night" and "See, do wide see" is a Scottish air, or possibly the melody of a Scottish song that Moore felt was unintelligible enough that he could put his own lyrics to it.

Bye
                                                                Mysha