If there is a significance to every time God changes someone's name in the Bible, what is the significance of his changing Saul's name to Paul?
That presupposes that God change Saul’s name. He didn’t. Scripture nowhere records God doing so, and we must resist the temptation to find a deep theological meaning of the sort found when God changes Abram’s name to Abraham, Jacob’s name to Israel, or Simon’s name to Peter.
The first century Mediterranean world was quite cosmopolitan, with people of different cultures and with different languages mixing freely. Consequently, people often went by more than one name, depending on the person or group they were conversing with.
We see the same phenomena today. When people from one language stock come to an area where a different language is spoken, they often choose a new name to use with their new community.
We see this in Scripture. The Evangelist we call Mark was named John at birth. John is a Jewish name (at least in its proper Hebrew form, Yochanan), but in non-Jewish circles he went by the Roman name Mark (Latin, Marcus; Greek, Marcos). In Scripture he is thus sometimes called "John whose other name was Mark" (Acts 12:12, 25). Sometimes he is called John (Acts 13:5, 13); other times he is just called Mark (Acts 15:39, 2 Tm 4:11).
Often, people moving into a new region pick names that are similar in some way to their birth name. For example, a Chinese person named Da-i who comes to live in America might begin going by the name David, because of the similarity of sound. In the same way, an American named James who comes to live in Mexico may begin going by Diego, because this is a Spanish form of the name James.
This appears to have been the case with Paul. What we see in the book of Acts is that, up until chapter 13, the apostle is referred to as Saul. Thereafter, except for flashbacks, he is referred to as Paul. Since Acts 13 is the chapter that Paul begins his apostolic ministry, which involves a great deal of international travel. He thus appears to have picked for himself the Greek name Paul (Paulos) to use in international, Greek-speaking circles. The likely reason for this is its similarity to his birth name, Saul (Heb. Sha’ul).
Just be glad that the figures of the Bible didn’t multiply names to themselves the way some of the Romans did. For example, the Emperor Claudius (Acts 11:28, 18:2) was named Tiberius Claudius Drusus Nero Germanicus and could properly be called by any one of those names (and not to be confused with the other first-century Roman statesmen Tiberius, Drusus, Nero, or Germanicus)!