There are two possible explanations for this name change.
1 - The first is found in the first book of Timothy when the Lord informs Saul that he would become an apostle to the Gentiles.
1 Timothy 2:7 "For which I was appointed a preacher and an apostle. I am speaking the truth in Christ, a teacher of the Gentiles in faith and truth."
Paul may have very well converted his Jewish name to a non-Jewish Gentile equivalent, so that the Gentiles would accept him more readily. At this time the Jews were intensely hated throughout the Roman Empire and very few Gentiles would listen to or be around a despised Jew.
2 - The second possibility may be that a new identity reflects a new life. After Paul met Jesus on the Damascus Road he was a new man with a new heart. The old man with the old beliefs were a thing of the past. Paul may have believed that this new man with a new heart required a new name.