Peter writes that Christians are elect just as Christ was elect and just as once Israel was the elect nation:

As you come to him, a living stone rejected by men but in the sight of God elect and precious, you yourselves like living stones are being built up as a spiritual house, to be a holy priesthood, to offer spiritual sacrifices acceptable to God through Jesus Christ. For it stands in Scripture:

“Behold, I am laying in Zion a stone,
a cornerstone elect and precious,
and whoever believes in him will not be put to shame.”

So the honor is for you who believe, but for those who do not believe,

“The stone that the builders rejected
has become the cornerstone,”

and

“A stone of stumbling,
and a rock of offense.”

They stumble because they disobey the word, as they were destined to do.

But you are an elect race, a royal priesthood, a holy nation, a people for his own possession, that you may proclaim the excellencies of him who called you out of darkness into his marvelous light. Once you were not a people, but now you are God’s people; once you had not received mercy, but now you have received mercy.

This last paragraph is an appropriation of Exodus 19. Peter is applying what was once the status of a nation to a trans-national institution:

Thus you shall say to the house of Jacob, and tell the people of Israel: You yourselves have seen what I did to the Egyptians, and how I bore you on eagles’ wings and brought you to myself. Now therefore, if you will indeed obey my voice and keep my covenant, you shall be my treasured possession among all peoples, for all the earth is mine; and you shall be to me a kingdom of priests and a holy nation. These are the words that you shall speak to the people of Israel.

Of course, Peter has already told the recipients of his letter that they are elect, in his greeting: “To those who are elect” (1 Peter 1.2). So just as Israel was a chosen nation and thus a nation of elect members, so now the Church is an elect institution and a body made up of elect members.

They have been chosen by God to be part of his family and kingdom.