Simon, son of Jona, was a fisherman of Galilee. It is by the name of Simon that Jesus talks to the apostle through the scriptures. There are three events where there are exceptions to the use of his name. Let's look at the three exceptions and a fourth reference to wrap it all up.
Andrew brought Simon to meet Jesus for the first time.
John 1:42 At that time Jesus said: Thou art Simon the son of Jona: thou shalt be called Cephas, which is by interpretation, A stone.
This is the Greek form of the Aramaic word for stone; not a small stone nor a big stone, just a stone. This is the very first time that Simon sees Jesus. He never heard him preach, teach or do any miracles before this moment. When Jesus saw Simon he gave him a prophetic name which would reflect a fulfillment of God's plan for him. Three future acts by Simon that would set him apart for a special ministry. There are only two times recorded in scripture where Jesus calls Simon by the name of Peter.
The first of two times that Jesus refers to Simon by the name of Peter.
Matthew 16.16 And Simon Peter answered and said, Thou art the Christ, the Son of the living God.
Understand that Simon saying that Jesus is the Christ is not the full revelation. Andrew knew that Jesus was the Messiah, the Christ, when he went to get Simon his brother. They followed Jesus because they believed he was the Christ. What set them apart from the rest of the people who would follow Jesus is the understanding that Jesus is also "the son of the living God."
Matthew 16.17 And Jesus answered and said unto him, Blessed art thou, Simon Barjona: for flesh and blood hath not revealed it unto thee, but my Father which is in heaven.
This is the revelation, that Jesus was the Son of the living God, that made the difference from all previous "Christs(?)" that came upon the scene during those days. It is the "blasphemy(?)" that gave the Sanhedrin its excuse to have Jesus crucified.
Another prophetic naming...
Matthew 16.18 And I say also unto thee, That thou art Peter (Petros), and upon this rock (petra) I will build my church; and the gates of hell shall not prevail against it.
At this point Jesus uses the Greek word for a small rock (Petros) and for a large rock (petra) instead of the Aramaic based Cephas (a stone). This was by design. It is the Petros (the man) declaring the petra (the foundational belief of the Christian Church). It is this belief that Jesus is the Christ and the Son of the living God that will stand against hell and gain salvation. Everything rests on these two points.
Promise to Peter...
Matthew 16.19 And I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven.
Here is his future ministry: binding and loosening. There is mention in scripture that whatever you forgive a man will be forgiven and that which you retain against a man will be retained (John 20.21-23). This is one example of using the keys of the kingdom. It is the most important of the uses. It is the ministry of forgiveness and restoration. The first unique opportunity to do this is done to Simon. It is the main ministry of the church.
Now we come to the night that Jesus is going to be betrayed. At the last supper Jesus prepares Simon for his part in the night and a future fulfillment of his promise.
Luke 22.31 And the Lord said, Simon, Simon, behold, Satan hath desired to have you, that he may sift you as wheat: 32 But I have prayed for thee, that thy faith fail not: and when thou art converted, strengthen thy brethren.
Then comes the second of two times Jesus refers to Simon with the Greek Petros.
Luke 22:33 And he said unto him, Lord, I am ready to go with thee, both into prison, and to death. 34And he said, I tell thee, Peter (Petros), the cock shall not crow this day, before that thou shalt thrice deny that thou knowest me.
We know that Simon did deny Jesus three times that evening. What was he denying? That he knew who Jesus was... thus denying that Jesus is the Christ, the Son of the Living God. He rejected the petra, the foundational belief of the church, out of fear. Bravery in a secluded field among friends was replaced by cowardice on a cold night among the enemies of Jesus. All eleven tasted this fear. Not one came forward as witness for Jesus. What torment of soul as they saw him suffer at the hands of his foes? What guilt they must have felt after they saw Jesus die on the cross?
That is the confession and denial of Peter.
We know that Jesus rose three days later. He visited the disciples Sunday evening. He was with them for many days. Yet things were still not quite right between Simon and Jesus. The next thing to happen is... Jesus restores Simon
John 21:15 So when they had dined, Jesus saith to Simon Peter, Simon, son of Jonas, lovest thou me more than these? He saith unto him, Yea, Lord; thou knowest that I love thee. He saith unto him, Feed my lambs. 16He saith to him again the second time, Simon, son of Jonas, lovest thou me? He saith unto him, Yea, Lord; thou knowest that I love thee. He saith unto him, Feed my sheep. 17He saith unto him the third time, Simon, son of Jonas, lovest thou me? Peter was grieved because he said unto him the third time, Lovest thou me? And he said unto him, Lord, thou knowest all things; thou knowest that I love thee. Jesus saith unto him, Feed my sheep.
Three times Simon denied knowing Jesus... three times broke faith with the "petra" revelation. It is at this time that Jesus gives Simon the opportunity three times to renew his confession. How does he do that? By asking Simon if he loves him. With each confession by Simon of his love for Jesus, Jesus tells Simon to feed my lambs. To feed those lambs with the truth... that Jesus is the Christ, the Son of the living God. Everything else in the gospels rest upon that truth.
Then come the fate-filled words that Simon longed to hear... follow me.
Though Jesus used the name Simon in friendly situations he used the name Peter when emphasizing the confession and again for the denial. It is amazing that Simon adopts the name Peter to keep the confession and denial foremost. That he is restored as Simon by Jesus afterward is the mark of forgiveness and restoration. It is Simon's testimony. It is his ministry: Acts 2, to the Jew first and Acts 10, to the Gentile world second.