Hi - a good place to start would be our origins thread on this song. You'll see at the top of this thread that I've crosslinked it to the other threads we have on this song. You'll also find some interesting comments on the song in this thread, but not that some of the information might be termed "apocryphal." Fred thinks the melody comes from the 18th century. I have a feeling he may be right, but that would destroy my romantic view of "Slane" as an ancient Celtic tune.
-Joe-

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Here's an excerpt from The Presbyterian Hymnal Companion (by LindaJo H. McKim, 1993):

339 Be Thou My Vision
Tune: SLANE

This hymn text is from an ancient Irish poem, "Rob tu mo bhoile, a Comdi cride" dating from around A.D. 700. The text was translated by Mary E. Byrne and included in the journal (vol. 2, 1905). The versification was by Eleanor Hull and included in the Poem Book of the Gael (1912).
Mary Elizabeth Byrne (1880-1931) was born in Ireland and graduated from the National University of Ireland in 1905, She was honored with the Chancellor's Gold Medal at the Royal University and was a member of the Royal Irish Academy. She was the coauthor of the Old and Mid-Irish Dictionary.
Eleanor Henrietta Hull (1860-1935), an Irish author and researcher, was founder of the Irish Text Society, The text has been altered for The Presbyterian Hymnal (1990).

SLANE is an ancient Irish ballad named for Slane, a hill near Tara, Ireland. In the fifth century, it was at Slane where the first fires of Easter were lighted by St. Patrick as a challenge to King Laoghaire. The tune was included in Patrick W. Joyce's Old Irish Folk Music and Songs (1901), where it was the setting for "With My Love Come on the Road."
The harmonization is by David Evans for the Revised Church Hymnary (1927).

I have to say that I had no knowledge of this hymn while I was growing up in the U.S. Catholic Church. I learned of it in the 1990s, through my contact with heathen folkies.
I also looked in the Companions to the Unitarian, Baptist, and Episcopalian (1940) hymnals, and they all say more-or-less the same thing as the Presbyterian Hymnal Companion. To my surprise, the 1940 Episcopalian hymnal did not have "Be Thou My Vision," but it did have two other hymns that use SLANE as melody. As far as I can tell, SLANE and "Be Thou My Vision" were not "married" until "Be Thou My Vision" first appeared in the revised Church Hymnary (Edinburgh, 1927).
"Be Thou My Vision" is described as an ancient Irish poem that "probably" dates to the eighth century, but its English translation (by Mary Byrne) was not published until 1905. SLANE is identified as a "traditional Irish air," but the earliest date of publication I could find was 1909, in Patrick W. Joyce's Old Irish Folk Music and Songs. And from all that, I think I can judge Fred's 18th century guess as correct, or perhaps even a bit early.
-Joe-