It is named after Cardinal Patrick Joseph Hayes, a previous archbishop of the Archdiocese of New York.

Cardinal Hayes' current rival is Mount Saint Michael Academy.

The two schools' football teams have met annually since 1942 on Thanksgiving Day.

Cardinal Hayes also takes part in non-annual football rivalries with Cardinal Spellman High School and Archbishop Stepinac High School for the Fathers' Club Trophy and the Father John Dubois Memorial Trophy, respectively.

Throughout the years, the school has been staffed by Archdiocesan Priests, De la Salle, Xavieran, Marist and Irish Christian Brothers.