Disputes on this topic are quite old.
St. John Chrysostom understands here that the rock upon which Christ builds His Church is the confession of the Apostle,
"'And I say unto you, You are Peter, and upon this rock will I build my Church;' that is, on the faith of his confession. Hereby He signifies that many were now on the point of believing, and raises his spirit, and makes him a shepherd. 'And the gates of hell shall not prevail against it.' 'And if not against it, much more not against me. So be not troubled because you are shortly to hear that I shall be betrayed and crucified.'" - St. John Chrysostom, Homily 54 on Matthew, 3
St. Augustine speaks of having in other places said Peter is the rock upon which the Church is built, but in his Retractions brings more clarity and invites the reader to consider whether it should be taken to mean Peter himself or Peter's confession and thus Christ Himself:
"I have said in a certain place of the Apostle Peter, that it was on him, as on a rock, that the Church was built. But I know that since that I have often explained these words of the Lord, Thou art Peter, and on this rock will I build my Church, as meaning upon Him whom Peter had confessed in the words, Thou art Christ, the Son of the living God; and so that Peter, taking his name from this rock, would represent the Church, which is built upon this rock. For it is not said to him, Thou art the rock, but, Thou art Peter. But the rock was Christ, whom because Simon thus confessed, as the whole Church confesses Him, he was named Peter. Let the reader choose whether of these two opinions seems to him the more probable." - St. Augustine of Hippo, Retractions 1.24 (quote here taken from the Catena Aurea of Thomas Aquinas)
Earlier we have in Tertulian's apologetic/polemic against Marcion, he writes,
"Was anything withheld from the knowledge of Peter, who is called 'the rock on which the church should be built,'" - Tertullian, Against Marcion, ch. 22
Origen in his Commentaries on John mentions the following in passing while discussing the writings left by the Apostles,
"And Peter, on whom the Church of Christ is built, against which the gates of hell shall not prevail left only one epistle of acknowledged genuineness." - Origen, Commentaries on John, 5.3
While in his Commentaries on Matthew the same Origen says,
"if we say it as Peter, not by flesh and blood revealing it unto us, but by the light from the Father in heaven shining in our heart, we too become as Peter, being pronounced blessed as he was, because that the grounds on which he was pronounced blessed apply also to us, by reason of the fact that flesh and blood have not revealed to us with regard to Jesus that He is Christ, the Son of the living God, but the Father in heaven, from the very heavens, that our citizenship may be in heaven, revealing to us the revelation which carries up to heaven those who take away every veil from the heart, and receive "the spirit of the wisdom and revelation" of God. And if we too have said like Peter, "You are the Christ, the Son of the living God," not as if flesh and blood had revealed it unto us, but by light from the Father in heaven having shone in our heart, we become a Peter, and to us there might be said by the Word, "You are Peter," etc. For a rock is every disciple of Christ of whom those drank who drank of the spiritual rock which followed them, and upon every such rock is built every word of the church, and the polity in accordance with it; for in each of the perfect, who have the combination of words and deeds and thoughts which fill up the blessedness, is the church built by God." - Origen, Commentaries on Matthew, 12.10
This is hardly exhaustive, but it does provide some insight into the ways the ancient fathers saw and understood the text.
-CryptoLutheran