I would like to suggest that "this rock" (Greek "Petra") references the apprehension and acceptance of that special revelation that Jesus is "the Christ, the son of the living God" (Mat 16:16); and this truth apprehended and accepted adds one to the "church" of which the second death has no authority to prevail over: "on this rock I will build My church, and the gates of Hades shall not prevail against it," Mat 16:18.
In Matthew 7:24, Jesus likened the man who heard and did his sayings to one who "built his house on the rock" (Greek "Petra"), and nature could not prevail against it, Mat 7.25. It was not just the sayings, but the apprehension and acceptance of them that was likened to a man building on the rock. Like in Matthew 16.18, that which is built on the rock will not be overcome by other forces.
It's also interesting that in Matthew 7.21-23, Jesus speaks of "never" knowing certain individuals who laid claim to doing certain things in the name of the Lord--these were obviously those who never apprehended and accepted the sayings of Christ or the revelation about Jesus and the Christ that Peter had apprehended and accepted.
The Church and its members are those who have "part in the first resurrection. Over such the second death has no power." Revelation 20:6 This could correspond to what Jesus said about the "gates of hell shall not prevail over it": hell is the second death, and this gates reference could be metaphoric of power or authority.
After Peter's confession that Jesus is "the Christ, the son of the living God," Jesus acknowledges the name of Peter: "you are Peter"; and then He says "upon this rock, I will build my church."
Why did Jesus call attention to Peter's name, which in Greek is "Petros," and then use the Greek word "petra" to reference this "rock" upon which he would build his church?
"Petros" is masculine in gender, while "Petra" is feminine. Is the gender difference significant? Do both words speak of a rock, but the masculine gender was used for "Peter" because he was a man? Maybe.
But maybe Jesus was making a contrast of sorts: Peter's was just a name, but the truth apprehended and proclaimed by Peter was the "rock" upon which the Church would be built.
Why didn't Jesus just say to Peter that he would build his church upon him--if Jesus was referring to Peter as the rock? Jesus could have said: "Peter, you are the rock upon which I will build my church." If Jesus did mean Peter, in what way did Jesus build his church upon him? Is there any evidence in the New Testament that Peter was meant as the rock?
Jesus does proceed to say to Peter that he will give "the keys of the kingdom" to "you": this "you" is in the singular. Whatever the "keys" are, Jesus is direct in saying to Peter that they will be given to him, but he wasn't similarly direct in calling Peter the "rock." Though the "keys" reference is directed to Peter, he speaks to the disciples in Matthew 18:18 about whatever "you [plural] bind on earth will be bound in heaven"--similar to what is said in 16:19. This just seems to speak of an authority they have: if they agree on something on earth, God will honor it ("I say to you that if two of you agree on earth concerning anything that they ask, it will be done for them by My Father in heaven. Matthew 18:19).
One can possibly find a role of Peter in the early stages of the "Church" in the book of Acts that stands out, but nothing in the New Testament beyond that gives a clue that would suggest Peter is the rock of which Jesus spoke.
Peter was the one who spoke on the day of Pentecost, Acts 2:14, and he proclaimed that Jesus was "both Lord and Christ." We are told in Acts 2:41 that 3000 "souls were added" to them, and "the Lord added to the church daily those who were being saved," Acts 2:47. It was those who believed the message and were saved that were "added" to the church. Peter may have spoke on Pentecost, but it was the truth he spoke by which people were saved and added to the church. The truth by which people are saved was not confined to Peter.
Peter had a confirmatory part in the response of the Samaritans, Acts 8.14-17. And then finally, Peter was sent to the gentiles, being Cornelius and company, in Acts 10, to "tell" them "words by which" they would "be saved," Acts 11:14.
Peter, after this significant role in the early stages of the Church, in the book of Acts, doesn't have as significant of a role as the Apostle Paul, when it came to the evangelism and growth of the more dominant gentile church. Peter is one of the "pillars," of the Jerusalem Church, as Paul said he "seemed" to be, Galatians 2:9. But one of those pillars was also James, the Lord's brother, who wasn't even an original Apostle, and James seemed to have a very prominent role in the Jerusalem Church. There is no indication that Peter held any significance in the Church other than his apostolic authority--which he shared with others. The only thing that stands out is his role in the early stages of the church in the book of Acts.
Jesus is the Christ, the son of God, and whoever believes in him for salvation will be saved. Some would say that revelation itself is the "rock," and that's understandable, considering that salvation comes by believing certain divine revelation, and whoever believes that revelation, is saved and added to the church. But considering the analogy of building on a rock in Matthew 7 about the man who listens to Christ's sayings, I think the rock could reference the apprehension and acceptance of the truth that Jesus is the Christ, the son of God.
Some will say that Jesus was referring to himself as "this rock." In Matthew 21:42, Jesus quotes the Psalm (118:22) : "Have you never read in the Scriptures: The stone which the builders rejected has become the chief cornerstone. This was the LORD's doing, and it is marvelous in our eyes'?
Jesus being the "chief cornerstone" is an argument for Jesus himself as the rock he was referencing. However, the word "stone" is not the Greek "petra" but "lithos." This stone is again referenced in Acts 4:11 and 1 Peter 2.7. The "household of God" is said to have Jesus as that cornerstone, Eph 2.20. Romans 9.33 and 1 Peter 2.8 speaks of Jesus as a stone and rock, not in a building sense, but as a cause of stumbling: a "stone of stumbling" and a "rock of offense." "Stone" is "lithos" which is masculine in gender and "rock is "petra" feminine in gender--if the gender proves anything. 1 Corinthians 10:4 speaks of the "spiritual rock" that followed the Jews with Moses, and that rock was Christ--the Greek word is again "petra (feminine in gender). So Jesus is clearly referred to as a stone and a rock, and though some usages are not about building but causing stumbling, he is the chief cornerstone.
So was Jesus referring to himself before Peter? He could have easily said that he was the rock upon which he would build his church. But he doesn't say anything like that. It was after Peter made the proclamation that Jesus said "upon this rock," which suggests to me that Jesus was referring to that proclamation, a proclamation of a specific divine revelation that Peter had apprehended--a revelation that if one apprehends and accepts, they will be added to the church.
