Then Saul, who was also called Paul, filled with the Holy Spirit, looked straight at Elymas and said... Acts 13:9

Jesus replied, “Blessed are you, Simon son of Jonah, for this was not revealed to you by man, but by my Father in heaven. And I tell you that you are Peter, and on this rock I will build my church, and the gates of Hades will not overcome it.  Matthew 16:17-18

As for Paul, Saul was his Jewish name.  Paul was his Latin/Roman name.  The Bible records his name as Saul of Tarsus up to his conversion.  After that the Bible calls him Paul.

Before his conversion, he was a Jew persecuting Jews (Saul).  After his conversion, he was a Roman citizen reaching out to Gentiles (Paul).  The name change seems to reflect his new purpose.  The name change emphasizes his new identity.

As for Peter, Simon was his birth name.  Peter was his nickname.  It means "rock."  We would say "Rocky."  He received that name because of his confession when asked by Jesus "Who do people say that I am?"  His answer: "You are the Christ, the son of the Living God."  Jesus liked that and said his confession was the rock-solid foundation that Jesus would build his church on.  The identity of Jesus.

Jesus calls him "The Rock Man"!

Both name changes speak of significant moments in a person's life.