Modern Popes, obviously, take special papal names upon their elevation to the papacy.
Peter, Clement, Antecletus, Linus and the earliest popes were all known by the given names, as were Somewhere along the line, I'm sure Wolseley knows when, it became common practice for Popes to being to refer to themselves by the name of a previous pope, taking their name as a papal name (there hasn't been an "originally" named pope, prior to JP1, since Pope Lucius in 1145). My guess would be it is done to represent the transition from the person who was just a bishop to the person who is now the Vicar of Christ, and ties into the monarchial practice of royalty taking "crown" names. Royals usually use their own given name as their reigning name, but they don't have to... Prince Charles could, if he wished, be King Frodo.
Often times this was done due to a certain feeling of affection for or connection to a predecessor. The pope before Karol Wojtyla took the unusual step of taking the names of his two immediate predecessors, John XXIII and Paul VI, who were the popes of the Second Vatican Council. This gave him the first ever compound papal name, Pope John Paul, and the first papal name in 800 years not followed by a Roman numeral.
Upon his untimely death, his successor, Wojtyla, took up his fallen predecessors name as Ionnes Paulus P.P. II, or Pope John Paul II.
It would be unthinkable at present for a modern pope to draw upon names other than the papal names of predecessor for his papal name. However, no pope would ever take the name Pope Peter II. Mainly due to superstition and fears regarding various Catholic prophecies about the end of the Church.