Why did Saul become Paul?

The name Saul is Hebrew, and the Greek version of that name is Paul. At a point in the book of Acts, Luke begins addressing Saul by his Greek name, Paul:

Acts 13:9 But Saul, who was also known as Paul, filled with the Holy Spirit, fixed his gaze on him, 

From this point forward in Acts, Luke uses only Paul’s Greek name, reflecting his assignment to preach the Gospel to the Gentiles. So Paul’s name was never changed, but Luke chooses to use his Greek name rather than his Hebrew name to reflect Paul’s mission to reach Gentiles with the Gospel. 

Scripture quotations taken from the (NASB®) New American Standard Bible®, Copyright © 1995, 2020 by The Lockman Foundation. Used by permission. All rights reserved. www.lockman.org