Why was Saul's name changed to Paul?
We aren't told. As Saul began teaching among the Gentiles, he switched his name from Saul to Paul. The transition is noted in Acts 13:9. Some speculate that Paul wanted to disassociate himself from his past when he persecuted the church. But this would only be a guess.
More likely it arises from the same reason Peter is also known as Cephas. Cephas is an Aramaic name for "rock" and Peter is the Greek word for "rock." Paul is a Latin word for "stop" or "pause." Saul is a Hebrew word for "inquire." Likely the name Paul was selected because it was close in sound to Saul. Paul stated, "For though I am free from all men, I have made myself a servant to all, that I might win the more; and to the Jews I became as a Jew, that I might win Jews; to those who are under the law, as under the law, that I might win those who are under the law; to those who are without law, as without law (not being without law toward God, but under law toward Christ), that I might win those who are without law; to the weak I became as weak, that I might win the weak. I have become all things to all men, that I might by all means save some" (I Corinthians 9:19-22). It is possible that Paul shifted his name from a recognizable Hebrew name to a Roman name to be better able to reach the Gentiles.