From Cory Schantz, an interesting article on The Gospel Coalition, emphasizing that Saul did not “change his name” to Paul upon becoming a Christian, but was always known by both names – “Saul” being the Hebrew form of it, and “Paul” the Greek.
When Saul/Paul launches his Gentile-focused ministry among primarily Greek-speakers (beginning with Acts 13:9), it’s natural for Luke, the author of Acts, to begin referring exclusively to him by his Greek name. Nor is it surprising that he’s later referred to as “Paul” in Jerusalem, since there were Greek speakers there too. Indeed, Luke could be making a thematic point by shifting from Saul to Paul around chapter 13, given the broader theme of Acts (e.g., 1:8). After all, the church’s nucleus is shifting from predominantly Jewish-centered Jerusalem to the Greek-centered “ends of the earth,” such as Rome.
I suppose that Jesus’ renaming “Simon” as “Peter” may have prompted a similar notion with Saul/Paul. Even the initial letters are the same! And we can make a similar pun as the one seen in Matthew 16:18, at least in Latin:
Tu es Paulus et super hoc paulum adificabit ecclesiam meam.
That is, “you are Paul, and on this little bit I will build my church” – something like the mustard seed in Matthew 13.