Saul’s Name was “not” changed to Paul after his conversion! This is a common fallacy which has been unintentionally perpetrated in our pulpits.
Saul was a Jew by birth and went by this name among the Jews. Paul was also a citizen of a Roman city, Tarsus in Cilicia; therefore, when he went around Romans, or Gentiles, he went by the name of Paul, a Roman name.
It was usual with the Jews to be called after this manner, that is, to have one name among themselves, and another among the Gentiles.
Observe in the Scriptures that Luke calls the Apostle by his Jewish name Saul, while he was among the Jews, and only preached among them. Yet because he was among the Gentiles, and was about to appear openly to be their apostle, he begins to use his Gentile name Paul!
S/N: It WAS customary with Jews to change a man’s name upon repentance and conversion; however, this is not the case or reason for why we see the use of Paul as opposed to Saul after his Damascus encounter. (Credit portions to the Commentary of John Gill)