I believe the original Greek name was Petras, which is literally translated "rock". If I am not mistaken, there is a famous natural landmark in Palestine that came to be called "Petra" because it is a giant rock formation.
I could be wrong about all this, though.
I will let my Roman Catholic brothers and sisters comment about the significance of this in terms of Catholic church polity and structure.
And you're partially correct. Jesus, or God, have been referred to as a "rock". The "rock" of our salvation and the "rock" in which we hide (referring to the Old Testament story of Moses being hidden in the cleft of the rock as God's glory passed by). Jesus is specifically called "cornerstone" and also, of particular interest to me, is the repeated statement that he is a "stone of stumbling".
I take this to apply to those who regard Him as a good teacher, a great man, a prophet, a revolutionary or an honorable example to follow...but cannot accept His own claim to be God in the flesh, a sacrifice for the sins of humankind and risen from the dead. Hence, they stumble over Jesus.
The origin is in the psalm 118 22-23. "The stone that builders rejected has now become the cornerstone [or "capstone"]. It is the Lord's doing and a marvelous thing to see."
I would take it to Jesus being rejected by the religious establishment. Capstone can refer to the absolute pinnacle of a building. In both contexts, the analogy is that Jesus is the completion . That analogy is also seen in First Peter 2:5, "you also, like living stones, are being built into a spiritual house to be a holy priesthood, offering spiritual sacrifices acceptable to God through Jesus Christ."
Christ compared himself with the physical Temple, in Mathew, Mark, and John, stating that he would tear it down and rebuild it in three days.
