The stone the builders rejected
Whom the rejected stone names first — Israel, David, or the coming king — and how the psalm was read liturgically.
Psalm 118:22-24
22The stone which the builders rejected has become the cornerstone.
23This is Yahweh’s doing. It is marvelous in our eyes.
24This is the day that Yahweh has made. We will rejoice and be glad in it!
What each tradition says
Rabbinic / Midrashic
Rashi identifies the stone that the builders rejected as a people. That people was humble among the peoples.
Rashi, Rashi on Psalms — The Judaica Press Tanach with Rashi, trans. A. J. Rosenberg (via Sefaria) — Rashi on Psalms 118:22
Patristic
Augustine states that the builders refused Him. Yet in the edifice the Lord was secretly raising, the stone they rejected was made the head stone of the corner.
Augustine, Expositions on the Book of Psalms, Psalm 118, §17 (NPNF ser. 1, vol. 8) — Exposition on Psalm 118, §17 (on ver. 22-23)
Catholic
Haydock’s note says the rejected stone was a sort of proverb applied to David, Zorobabel, or the Jewish nation. These can only be figures of Christ, in whom the prediction was fulfilled when he established his Church and made one people of those previously divided.
George Leo Haydock, George Haydock's Catholic Bible Commentary (1859) — Note on Psalm 117:22 (Douay; = Ps 118:22, 'the stone which the builders rejected')
Orthodox
Chrysostom says that by the stone He means Himself, and by the builders the teachers of the Jews. He cites Ezekiel’s words: “They that build the wall, and daub it with untempered mortar.”
John Chrysostom, Homilies on the Gospel of St. Matthew, Homily LXVIII (NPNF ser. 1, vol. 10, trans. George Prevost) — Homily LXVIII, on Matt 21:42 (the stone which the builders rejected, Ps 118:22)
Reformed / Protestant
Calvin states that being abandoned by the chief men is of little importance, since he was visibly chosen by God's judgment to be king over Israel. He compares himself to a stone, and the principal rulers of the Church to master-builders.
John Calvin, Commentary on the Book of Psalms, vol. 4 (trans. James Anderson; Calvin Translation Society) — Commentary on Psalm 118:22 ('The stone which the builders rejected')
Evangelical
Wesley says the commonwealth of Israel and the church of God are compared to a building, in which the people are the stones and the princes and rulers are the builders. He says these master-builders rejected David, and their successors likewise rejected Christ.
John Wesley, Explanatory Notes upon the Old Testament — Notes on Psalm 118:22