Why is only Peter mentioned out of the twelve? Why is he singled out? It is because Jesus thrust upon Peter the responsibility of strengthening the other disciples after he repented. We have to understand that Satan desired to sift them all, but Jesus only addresses Peter, giving him the responsibility to buck everybody else up.
Why Peter? Why did He not say, "John, you son of thunder, I want you to be a rod of lightning to these men and encourage them after these things happen." No, he says this to Peter. He does not say to John, "I want to you to work with your brother and strengthen him." He says it to Peter because Jesus had given him the responsibility to be the bulwark of strength (humanly speaking) amongst the Twelve.
John W. Ritenbaugh
Unity (Part 2): God's Pattern of Leadership
Jesus asks His Father to strengthen Peter's faith. But notice Jesus' perception: He could see that Peter would stumble—and stumble very badly. He had such confidence that God would turn Peter around that He says, "When you return, when you are converted and come back to Me, strengthen your brethren." That is how confident Jesus was that God would hear His prayers.
Faith is the foundation of Christian character. Without it, we have no access to God. "He who comes to God must believe that He is" (Hebrews 11:6). Satan was out to destroy Peter's trust in God, and Jesus acted to guard him. "That your faith should not fail" means that it should not come to an end or disappear completely. Peter did stumble badly, but he also got up and went on. It is entirely possible that Peter had a lot of confidence (remember that he says in verse 33, "Lord, I am ready to go with You, both to prison and to death"), but that confidence was in himself.
For Peter to have the right kind of faith, God had almost to smash the man where he could easily see his faith, his confidence in himself, as absolutely nothing, and that if he were going to have a good relationship with God, it would have to be on the basis of his confidence in God, not in Peter. Peter had to stumble in order to have true faith in God. When faith is broken down, the foundations of true spiritual life give away. That is why Satan wanted to destroy Peter's faith. If he could do that, the entire structure of the man's relationship with God would collapse, but God did not allow it.
John W. Ritenbaugh
Faith and Prayer