All popes up until 533 used their real names, although there were already some duplicates - the first one whose name had already been the name of a previous pope was Sixtus II (257-258), and by the 6th century there was already a Felix IV (526-530).

Felix IV was succeeded by his personal choice, Boniface II, but that wasn’t the usual procedure, so another candidate, Dioscorus, was also elected (although he died after a month). Boniface also tried to force the church to accept his own hand-picked successor, but the church disagreed and he did not get his way. He died in 532, and some months later, John II was elected, so there may have been some unrecorded difficulty with his election as well.

He may have wanted to break with tradition after the unusual circumstances of the previous few years. But more importantly, his real name was Mercurius, the name of a Roman god. His father’s name was Proiectus. Presumably they were a Christian family but maybe they had been pagan within living memory? Whatever the case, Mercurius served as a priest in the church of St. Clement on the Caelian hill in Rome and his name caused no problems there, but evidently he thought “Pope Mercury” would be in appropriate, so he changed his name to John. It’s unknown if he had a particular John in mind but he might have been honouring Pope John I, who was arrested by the Ostrogoth king Theoderic and died in prison a few years earlier in 526 (and was succeeded by Felix IV). If so, this might have been a direct slight against the Ostrogoths, and maybe against the memory of Boniface II (who may have also been an Ostrogoth).

The next pope to change his name was the next John, John III, born Catelinus (561-574). Then everyone used their real names for several hundred years until Octavian, who became John XII (955-964) and John XIV (983-984, born Peter Campanora). You can see a pattern - popes whose names sounded too much like a pagan Roman name felt that they should change it (Mercurius, Catelinus, Octavianus). Peter Campanora had a good Christian name but since Peter was considered the first Pope, and there has never been a Peter II - at that point it was simply because no one else named Peter had ever become Pope, but by then it would have seemed strange to use “Peter II”, so he changed his name too. Everyone who had changed their name so far had changed it to John, the most common papal name (then and now).

Changing names became the norm in 996 with Bruno of Carinthia, who took the name Gregory V (996-999) and Gerbert of Aurillac, who became Silvester II (999-1003) - notably they were the first German and French popes, respectively, with unusual names compared to all the Latin and Greek-named popes that came before. Pope Bruno? That would be too strange. I’m not sure why Bruno chose the name Gregory, except obviously in honour of Gregory I the Great. Gerbert of Aurillac chose the name Sylvester in honour of Silvester I, who was pope in the time of Emperor Constantine in the 4th century. Gerbert was the tutor of Holy Roman Emperor Otto III, who liked to be seen as a new Constantine, so Gerbert probably wanted to claim that he and Otto had that same sort of relationship.

Almost everyone afterwards has changed their name, except Marcellus II, who was pope briefly in 1555. There’s no reason why they couldn’t have used their real names; it just became a tradition that they didn’t. There’s also no reason why a new pope can’t choose an entirely new name that has never been used before, which is what happened with John Paul I (in honour of the preceding popes John XXIII and Paul VI) and Francis (in honour of Francis of Assisi).

Source for the early popes:

John Moorhead, The Popes and the Church of Rome in Late Antiquity (Routledge, 2015)