This refers to the confession Peter made, not Peter. The name Peter means "pebble". The "rock" is a different word.
Actually, Jesus spoke Aramaic, and, as John 1:42 tells us, in everyday life he actually referred to Peter as Kepha or Cephas (depending on how it is transliterated). It is that term which is then translated into Greek as petros. Thus, what Jesus actually said to Peter in Aramaic was: "You are Kepha and on this very kepha I will build my Church."
The Church Fathers, those Christians closest to the apostles in time, culture, and theological background, clearly understood that Jesus promised to build the Church on Peter, as the following passages show.
Tatian the Syrian
"Simon Cephas answered and said, ‘You are the Messiah, the Son of the living God.’ Jesus answered and said unto him, ‘Blessed are you, Simon, son of Jonah: flesh and blood has not revealed it unto thee, but my Father which is in heaven. And I say unto thee also, that you are Cephas, and on this rock will I build my Church; and the gates of hades shall not prevail against it" (The Diatesseron 23 [A.D. 170]).
Tertullian
"Was anything withheld from the knowledge of Peter, who is called ‘the rock on which the Church would be built’ [Matt. 16:18] with the power of ‘loosing and binding in heaven and on earth’ [Matt. 16:19]?" (Demurrer Against the Heretics 22 [A.D. 200]).
"[T]he Lord said to Peter, ‘On this rock I will build my Church, I have given you the keys of the kingdom of heaven [and] whatever you shall have bound or loosed on earth will be bound or loosed in heaven’ [Matt. 16:18–19]. . . . What kind of man are you, subverting and changing what was the manifest intent of the Lord when he conferred this personally upon Peter? Upon you, he says, I will build my Church; and I will give to you the keys" (Modesty 21:9–10 [A.D. 220]).
The Letter of Clement to James
"Be it known to you, my lord, that Simon [Peter], who, for the sake of the true faith, and the most sure foundation of his doctrine, was set apart to be the foundation of the Church, and for this end was by Jesus himself, with his truthful mouth, named Peter" (Letter of Clement to James 2 [A.D. 221]).
Good grief...