Pope John II (r. 533-535) was the first to adopt a regnal name, because his given name was "Mercurius" after the Roman god Mercury. The story goes he felt it improper for a Pope to beat the name of a pagan deity so chose the name John instead.

This didn't standardize the practice, however, and popes continued to use their personal names into the Middle Ages and beyond. I recall reading somewhere that it became more common during the Carolingian era and beyond because many popes by then had Germanic names (e.g. Hildebrand). Choosing a more traditionally "Roman" name would allow a foreigner to fit in better in Rome. Maybe someone else can elaborate.

It's not actually a requirement even today, but merely custom. The last pope to simply keep using his given name was Marcellus II (r.1555).