Has anyone ever heard of a song by this title? I’m thinking it was a 19th- or early 20th-century, sing-around-the-piano-type song. I can’t find it in any of the Internet search engines, and I would SWEAR that it’s a real song.
You are probably thinking of “My Wild Irish Rose” by Chauncey Olcott (who also wrote “When Irish Eyes Are Smiling”), published 1899 for the play A Romance of Athlone. You can find the lyrics at http://www.contemplator.com/ireland/wildrose.html.
There is also “Ramblin’ Rose,” written by Joe Sherman and Noel Sherman, recorded by Nat King Cole, lyrics at http://www.contemplator.com/ireland/wildrose.html; and “Rambling Rose,” from 1948, music by Joseph Burke and lyrics by Joseph Allan McCarthy, recorded (separately) by Perry Como and Dean Martin, lyrics at http://lyricsplayground.com/alpha/songs/r/ramblingrose.html.