After the counting of the ballots produces a consensus candidate, the
new pope is asked, “By which name will you be known?” The new pope then simply states what his name will be.
Occasionally, he may offer an explanation for his choice, but no one expects that.
Sometimes, he may offer an explanation later.
John Paul I explained his unusual choice of a double name by saying he intended to carry on the legacy of both John XXIII and Paul VI.
After his brief pontificate of just 33 days, the new pope took the name John Paul II, he said to provide continuity in respecting the wishes of his short-lived predecessor.
It was widely assumed Benedict XVI chose his name to honor the legacy of Benedict XV and/or St. Benedict.
The first pope to change his name did so in the Sixth Century because he was named after the Roman god Mercury, and he thought it inappropriate to carry that name as pope.
He chose to be called John II.
After that some popes changed their names and some did not.
The last pope to use his baptismal name was Marcellus II in 1555. Since then, it has been customary for all popes to change their names, although it is not required in any way.
Some say that it would be a welcome change for popes to be called by their baptismal name.
This would be more in line with the famous saying by St. Augustine that underlined the importance of baptism over holy orders: “I am fearful of what I am for you, but I draw strength from what I am with you. For you I am a bishop, and with you I am a Christian. The former designates an office received, the latter the foundation of salvation.”