When Archbishop Dolan takes the reins of New York on the Wednesday in the Easter Octave, Edward Cardinal Egan will become the first and (so far) the only "former" Archbishop of New York. All of his predecessors, all eight Archbishops (O'Connor, Cooke, Spellman, Hayes, Farley, Corrigan, McCloskey, and Hughes) and three Bishops (Dubois, Connolly, and Concanen) died in office.
And, since Egan will still be alive, what this also means is that, when [yes, I know that's an "if and when"] Dolan is created a Cardinal, for the first time in over half a century, the Minor Basilica of Saints John and Paul will not be available. Cardinals keep their titular for life, even in retirement from "active ministry". The only exception being if they are promoted within the College of Cardinals, in which case they receive another titular church from the Holy Father.
Look at the case of Cardinal Law, who was named titular of Rome's Santa Susanna when he received the red biretta as Archbishop of Boston in 1985. Like Sts. John & Paul, Santa Susanna had a similar affinity with the Archdiocese of Boston, having been the titular of Boston's Cardinals Cushing (1958-1970) and Medieros (1973 to 1983). Even when Cardinal Law was named Archpriest of Santa Maria Maggiore in 2004, he kept Santa Susanna as his titular church, something which continues until today. What that meant was, when Cardinal O'Malley was elevated in 2006, Santa Susanna was not available to him. Amazingly, the church right across the street from Santa Susanna was, and so Cardinal O'Malley is the titular of Santa Maria della Vittoria.
But I digress. When Archbishop Francis Spellman was created a Cardinal in 1946, Pope Pius XII showed his affection for Spelly by naming him the titular of the church of Ss. Giovanni e Paolo, which he himself (as Cardinal Pacelli) had served as honorary pastor/protector. Since then, the church on Rome's Celian hill has become part of the patrimony of the New York Archdiocese. Cardinal Spellman paid for the restoration of the church's facade and portico as well as the costs of opening the excavations underneath the church to the public (which allows one to visit the ruins of the house in which Saints John and Paul, officers of the Emperor Julian, lived, were killed, and buried).
If they want to go farther back in history, there are a two other places they could choose. From 1924-38, Patrick Cardinal Hayes' titular was Santa Maria in Via. From 1911-18, John Cardinal Farley's titular was Santa Maria Sopra Minerva, which was also the titular of his predecessor, John Cardinal McCloskey, who served Sopra Minerva from 1875 to 1885.
Even though he'll soon be relieved of the day to day operations of the New York Archdiocese, Cardinal Egan still serves as a member of the Vatican's Congregation for Eastern Churches, the Supreme Tribunal of the Apostolic Signatura, the Pontifical Council for the Family, the Prefecture for the Economic Affairs of the Holy See, and the Council of Cardinals for the Study of Organizational and Economic Affairs of the Holy See.