(apologies to Carl Weathers)
Peter (Petros, a chunk of rock) was originally named Simon (or Simeon, meaning "hearing"), a common Hebrew name. At some point Jesus changed his name.
In Matthew, Simon is referred to as Peter fairly early (starting from 4:18), although nowhere does Jesus specifically address him as Peter until 16:18 (right after Peter's proclamation). Five verses later, Jesus refers to Simon as "Satan" (v23).
Mark 3:16 is "And Simon he surnamed Peter", although this may be a comment (rather than an event in the timeline). Peter's proclamation is in 8:29. Interestingly, Jesus only refers to Peter as "Satan" (Mark 8:33) and "Simon" (Mark 14:37) (at least, when searching for Peter).
In Luke, Jesus does not address him as Peter until 22:34 (where Jesus predicts he will deny Him). The proclamation is in 9:20.
In John, Jesus addresses him as Simon (John 21:15).
That Jesus refers to building upon the proclamation, rather than the person, should be clear:
Matthew 16:18 "And I say also unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it."
"Peter" is petros (chunk of rock, as above). "This rock" is petra, meaning "mass of rock" (there is a city built into a mountain referred to as Petra near Israel).
This interpretation is hinted at in Catechism 442: "From the beginning this acknowledgment of Christ's divine sonship will be the center of the apostolic faith, first professed by Peter as the Church's foundation."