A few of my favorites which may or may not fit your category:
Roddy McCorley
Shule Aroon (or any of its various titles and tune variants)
Mo' Mary (Irish or Scots?)
Mick McGuire
Finnegan's Wake
Bridget O'Malley


"The Mountains O' Mourne" is a Percy French song (at least the lyrics are) but definitely one of my favorites.