Hello! "Paul," or Paulus, was a common Roman cognomen (e.g., L. Aemilius Paulus, the conqueror of Macedonia). Since Paul was born a Roman citizen, his family may have adopted the cognomen of their benefactor. The late Colin Hemer wrote a short article on this topic, if you're interested ("The Name of Paul"). :)

Edit: Forgot to point out that Paul, by adopting his Roman name, might have wanted to emphasize his "Roman-ness" to a wider Mediterranean audience.