The origin of this tune is shrouded in mystery, although there are several theories (as is the case whenever historians get involved in anything). I first knew it as Danny Boy, but those lyrics are 20th century. Then in Mingo Glee Club we learned the traditional words under the title “Londonderry Air”. Apparently, it first appeared in print in 1855, after being submitted by Miss Jane Ross, who was taking down traditional Irish folk tunes and heard this one from an itinerant fiddler in the county of Londonderry.
Go here for a more complete account.
These pages are from the Treasure Chest of World-Wide Songs, published in 1936, Treasure Chest Publications, New York, New York.