Peter is quoting from Isaiah 28:16. Rulers in Jerusalem had agreed that they would do whatever it took to avoid death and so build their future (Isaiah 28:14–15). In the face of their plans for a future, the Lord reveals his plan, namely, that he will lay a stone that will serve as a solid foundation for the future. And he called on all to believe the stone. By quoting this passage in his letter to the exiles, Peter is saying that in Jesus Christ God is now doing what he long ago said he would do. Describing Jesus as “a living stone” is simply a repeat of Old Testament Scripture!
6 For it stands in Scripture: “Behold, I am laying in Zion a stone, a cornerstone chosen and precious, and whoever believes in him will not be put to shame.”