As the comments below highlighted to me, it is possible that Paul didn’t change his name. Rather, as it says in Acts 13:9, Saul ‘was also called Paul’ — i.e. he (always had) had two names. Referring to the Saul/Paul, F. F. Bruce says:
He was born into an orthodox Jewish family, and as his father was a Roman citizen, he inherited this distinction, a rare one among eastern Jews. In Jewish circles he bore the name of Saul, but in the Gentile world he was commonly known by his Roman cognomen Paullus, Anglicized as Paul […]
Saul’s father was a Roman citizen… [so] Saul was born a Roman citizen… As a Roman citizen, he had a Roman name, consisting of three parts: praenomen, nomen gentile and cognomen.2 What his praenomen (first or personal name) and nomen gentile (family name) were we can only guess; his cognomen, however, was Paullus, by which (in its English form Paul) we usually call him.
In family circles, however, he was known by his Jewish name Saul.
2. The threefold Roman name may be illustrated from the full names of such famous Romans as Gaius Julius caesar, Marcus Tullius Cicero, Lucius Cornelius Scipio. …
— F. F. Bruce, The Spreading Flame (Exeter: Paternoster Press, 1958), p. 18, 81
However, even if Saul/Paul didn’t change his name, after Acts 13:9 the Biblical record, as one commenter points out below, definitely shifts from referring to him as Saul and, instead, refers to him as Paul. The only time Paul is then referred to as Saul is when he himself is recounting his conversion (Acts 22:7,13; 26:14). We could legitimately ask why this shift happens.
Although we aren’t told why there was a shift, here is a possibility:
Saul was also the name of the first human king of Israel (God was the first king: 1Sam. 8:7); that Saul was very tall (1Sam. 9:2) and he also became a very bad king (1Sam. 15). King Saul was a good king when he was ‘small in his own eyes’ (1Sam. 15:17) — i.e. when he was a humble person. When he became proud he became a wicked king.
In contrast, Paul means little or small — perhaps Paul himself instigated the shift from being referred to as Saul to being referred to as Paul because once he had been a bad person, proud (Phil. 3:4-6), like king Saul, but now he was a Christian trying to be humble (Phil. 3:7-11). Perhaps he saw ‘Paul’ now as the more fitting of his names.
Another possibility (as suggested in a comment below) is that Paul chose ‘Paul’ over ‘Saul’ because he was, by and large, a preacher to gentiles (i.e. non-Jews; Rom. 11:13), so his gentile name would possibly be more preferable in such circumstances.