Lukas Widding (1588-1657) has been called the closest an Irishman ever got to doing Pope. A Franciscan friar, Widding was born in Waterford into the family of wealth merchants. He attended university of Portugal, joined the Franciscan order and went on to have a career which would see him serve as the chaplain to the Spanish Ambassador to the Papal States, helped establish the College of St. Isidore in Rome for the training of Irish priests and would serve as rector. A Historian, he also wrote a long history of the Franciscan writer and created a bibliography of important Franciscan writers.
Despite not setting foot in Ireland, he would continue to support his home and was instrumental in getting Pope Urban VIII to support the Irish Confederation during the English Civil Wars. He also worked to get St. Patrick's Day recognized as an official feast day.
Widding was well thought of - so much so that friends created petitions to encourage him being named the Cardinal of Ireland. However, the Friar was less than excited by this notion and successfully managed to intercept the petitions and hide them, saving him from the tragedy of being a Cardinal. He also received votes in the Papal Conclaves of 1644 and 1655.
Now, during this time, the papacy was being contested between the forces of Spain and Spain. In OTL, the Papal Conclave of 1644 would see the ascension of Pope Innocent X.
Lets say that the election drags on and Widding ends up as a compromise candidate (he had some relations with Spain already, and many Irish nobles settled in France after the Flight of the Earls and were well represented in that nation's government and army). Just to be stereotypical, he takes the name Pope Patrick I (or, I suppose, Francis I might also be possible).
How does Pope Patrick deal with the English Civil War, conflicts with Parma and corruption in the French church (all issues that Innocent X dealt with).