The Only Person for Whom Jesus Prayed in the Gospels

Taken from "Catholic Apologetics" by Fr. John Laux:

"And the Lord said: Simon, Simon, behold Satan hath desired to have you that he may sift you as wheat. But I have prayed for thee that thy faith fail not, and thou being once converted confirm thy brethren" (Luke 22:31-32). Christ's prayer - the prayer of God - was necessarily effective; by it Peter was permanently confirmed in the faith which never failed him, though his moral courage did. Christ prayed for all His Apostles (John 17:9ff); but He prayed in an especial manner for Peter, upon whom the duty devolved of "strengthening his brethren." It is the only record we have of Our Lord offering prayer for an individual. This fact alone helps us to define Peter's relation to his fellow-Apostles. 

Three times the risen Lord asked Peter: "Lovest thou Me?" And after Peter's threefold assurance: "Lord, thou knowest that I love Thee," the Good Shepherd makes Peter shepherd in succession to Himself. "Feed My lambs, feed My sheep" (John 21:15-17) 

There can be no doubt as to the meaning of Christ's words. In the language of the Old Testament (cf. Ezech. 37:22-25) as well as in the Greek language, kings are rulers are called shepherds of the people. To Peter is given the office of leader and ruler of Christ's people. He is not primus inter pares - "the first among his peers," as the Anglicans and Episcopalians would have us believe, but he is their Shepherd, their guide and leader, their Supreme Head.

More information: Primacy of Peter