As one who was a Roman Citizen by birth, he would have had three names. the first name is the praenomen [which was a formal name]. The second was the nomen which was the tribe one belonged to. The nomen could be one's own tribe or the tribe of one's patron. The third name was the cognomen. This would be equivalent of our first name. Paul is a Latin name. Even though he used Greek, he was still influenced by Roman/Latin.
Tarsus where he was born, had at some point patrons such as Julius Ceasar or Marc Antony. If his family got their citizenship from either of these two and thus took on the names of their patrons, then they'd pick either Gaius Julius Paulus or Marcus Antonius Paulus, or the like [if someone else had been the patron to grant citizenship]. Or, some have noticed that a local tribe in Tarsus called the Aemilius tribe had a lot of Paulus in it. So he could have had as his name L. Aemilius Paulus. The book doesn't explain the L.
The book is "Paul and his letters" by John Polhill.