He was born on 26 May 1916 in County Leitrim, and was named after Joseph Plunkett, one of the leaders of the 1916 Easter Rising. His father Andrew Mooney (born Kilteevan, County Roscommon) married Josephine Gaffney of Drumshanbo in New York in 1908, and relocated to Drumshanbo, County Leitrim where he ran the family business of bar, grocery and drapery shop. Elected a Sinn Féin Councillor in 1919, Andrew Mooney was interned in Ballykinlar Camp, County Down by the British forces during the War of Independence. Following his release in late 1921 he took the pro-Treaty side joining Cumann na nGaedheal, but left that party over a Dáil nomination dispute in 1929. He remained an independent councillor until his sudden death in 1943.