There are certain circles in which it is commonly thought and taught that Saul was Paul's Jewish or Aramaic name and Paul was his Greek name. Thus, when Paul became the apostle to the Gentiles, Acts uses the Greek Gentile version of the name (Paul) and the Jewish version (Saul) then disappears. The situation is actually a bit more complicated, as a brief perusal through NT surveys quickly shows.
In their recent publication The New Testament in Antiquity (Zondervan, 2009), Wheaton College faculty Gary Burge, Lynn Cohick, and Gene Green explain the Saul-Paul name question as follows: 'Paul was a Jew and also a Roman citizen..., born in the free city of Tarsus, the capital of the province of Cilicia. As a Roman Paul would have had three names: praenomen, nomen, and cognomen. A Roman's cognomen acted as a surname and, in the apostle's case, this was the Latin Paullus (Gk. Paulos), which identified him as a member of the Paulli family. He was known also as Saul (Gk. Saulos, a transliteration of the Semitic Shaul.... The name was likely his supernomen, a kind of nickname used chiefly with Jews' (p. 251).
Robert Gundry in his Survey of the New Testament (4th ed.; Zondervan, 2003) does not speculate on where the name Saul comes from, but he is also of the opinion that Paul is the surname. Gundry states that Paul's praenomen and nomen 'have not survived' (pp. 312-13).