There is no usage of "petros" as small rock in patristic Greek. It is a rare word. "Petra" is rock, and Petros is simply a masculine form of "petra". It is clear form the passage that Jesus praised Peter's faith and so renamed him after the Rock on which He will build His church.
Indeed, that does not make Jesus Himself any less of a "rock of our salvation". What is shows is that St. Peter was chosen as some kind of a human fundament for the Church. That is what the Papacy is.
1 Peter 2:5, by the way, uses another word, "lithoi".
For those who have not read, or have forgotten the Scripture:
JOHN 1:
40 One of the two who heard John speak, and followed him, was Andrew, Simon Peter's brother.
41 He first found his brother Simon, and said to him, "We have found the Messiah" (which means Christ).
42 He brought him to Jesus. Jesus looked at him, and said, "So you are Simon the son of John? You shall be called Cephas" (which means Peter).
Jesus had already named him "Peter" thus could not have re-named unless, of course, you are thinking of:
JOHN 21:
15 When they had finished breakfast, Jesus said to Simon Peter, "Simon, son of John, do you love me more than these?" He said to him, "Yes, Lord; you know that I love you." He said to him, "Feed my lambs."
16 A second time he said to him, "Simon, son of John, do you love me?" He said to him, "Yes, Lord; you know that I love you." He said to him, "Tend my sheep."
17 He said to him the third time, "Simon, son of John, do you love me?" Peter was grieved because he said to him the third time, "Do you love me?" And he said to him, "Lord, you know everything; you know that I love you." Jesus said to him, "Feed my sheep.
Could it be that Jesus took Peter's "rockiness" away?