Moore's The Minstrel Boy was set to a much older tune called Mórín, or The Moreen.  It was printed in Charles Villiers Stanford's edition of George Petrie's Complete Collection of Irish Music (vol. 3, 1905-ish) as no. 1067; Petrie got it from the O'Neill MS. of 1787.  The tune is available at Bruce Olson's  Roots of Folk:

ABC's of Miscellaneous 18th Century English, Scots and Irish Tunes

Alternatively, the same file may be accessed through  JC's ABC tune finder,  (search for moreen) where you will also have the option of hearing it as midi or seeing staff notation.