Answer:
This is from one of my commentaries that I use:
Ac 13:9 Then Saul . . . also . . . called Paul–and henceforward Paul only; a softening of his former name, in accommodation to Roman ears, and (as the word signifies “little”) probably with allusion as elsewhere to his insignificance of stature and appearance (2Co 10:1,10)
Others say it was just that is was his “Gentile name” as compared to the Hebrew name of Saul. I however, believe that he chose this name because it signified being little, small or weak where in comparison Saul was a name associated with great stature.
After all, it is Paul who wrote 2 Corinthians 12:10 Therefore I take pleasure in infirmities, in reproaches, in necessities, in persecutions, in distresses for Christ’s sake: for when I am weak, then am I strong.