Saul is שָׁאוּל in Hebrew ("begged/asked God") and Σαῦλος is the translation of the LXX. שָׁאוּל is the name of the first king of Israel, of the tribe of Benjamin, and Παῦλος is the Hellenization of the Latin name Paulus, which literally means "small." Research has made many assumptions to answer the question of the name change. It has been asserted that the Latin name of the apostle was Gaius Julius Paulus, because the family of the apostle––presumably receiving Roman citizenship after the birth of the child Saul––had adopted the name of the famous family to which the general Emilius Paulus belonged. The other two words, Gaius Julius, would have been given to the child by his parents in honor of Julius Caesar, a person from whom the Jews had greatly benefited. But this hypothesis has no basis in the texts that exist today. Before we move forward with the question at hand, it might be necessary to focus our attention for a moment on another: How was a Roman name actually formed?
A Roman name had three parts. Take, for example, the name Marcus Tullius Cicero. The first part was is praenomen (i.e., that which is in front of the name). E.g,. Gaius, Lucius, Marcus . . . This corresponds to what in the present day we call the name of each person. The second part was the nomen (i.e., name or "demonym"). This is the designation as the "gens," tribe or clan, to which each individual belonged. Each citizen received as part of his name, the "demonym." In our example, Tullius. This person was therefore a descendent of the tribe, or gens, of Tullia. In a broad sense, this is like the "last name." The third and final part was the cognomen or the specific designation––sometimes a nickname. This is the name used by a person's family or tribe. In the present example, it's Cicero, which literally means that was called specifically to a "family" within each gens or tribe. In our case, it's Cicero, which literally means "chickpea."
So, in the letters that we know to have been written by the apostle, we only find the name Paul (Παῦλος). The Greek sounds Latin. In the change from Σαῦλος to Παῦλος, there is a play on words. Only one phoneme is changed. That's all. It's quite a curious change. One is the name of a monarch of Israel, whom tradition portrays as great and handsome, the other a name meaning "small." Why?