I think that part of the reason why elected popes take a new name is Biblical tradition: new names are given to persons whom God has entrusted with a new mission or an honor of great magnitude.
Examples:
Abram became Abraham to signify that he would be the father to many peoples, after he manifested his willingness to sacrifice his one and only son, Isaac.
Jacob (Abraham’s grandson) became known as Israel, to signify the nation comprised of the 12 tribes that started with his sons.
Jesus renamed Simon as Peter (rock) to signify that Peter would be the rock of the Church Jesus would found through him.
Saul became Paul after he was knocked off his horse on the road to Damascus and converted from a Christian-killer to the great evangelist that wrote most of the letters in the New Testament.
Every Catholic who is receives the sacrament of Confirmation takes a new name to signify the renewal of his/her baptsimal vows and the blessings received from the
infusion of gifts from the Holy Spirit.