Yahweh's regret
'Yahweh regretted that he had made Saul king' beside 'the Glory of Israel will not lie nor repent' — divine mutability and accommodation (also Genesis 6:6; Jonah 3:10).
1 Samuel 15:10-11
10Then Yahweh’s word came to Samuel, saying,
11“It grieves me that I have set up Saul to be king; for he has turned back from following me, and has not performed my commandments.” Samuel was angry; and he cried to Yahweh all night.
1 Samuel 15:29
29Also the Strength of Israel will not lie nor repent; for he is not a man, that he should repent.”
What each tradition says
Rabbinic / Midrashic
Rabbi Yehuda says God's regret was that He created man below, for had He created him on High he would never have rebelled. Rabbi Neḥemya says God takes solace in having created him below, since on High he would have incited the supernal beings to rebel as he did the lower beings.
Bereshit Rabbah (Genesis Rabbah; anonymous midrash); English: The Sefaria Midrash Rabbah, 2022, Bereshit Rabbah (Genesis Rabbah) — Bereshit Rabbah 27:4, on Genesis 6:6 ('The Lord regretted that He had made man')
Patristic
Tertullian says that in Saul’s case the Creator made no mistake in selecting him for the kingdom and endowing him with His Holy Spirit. God most fitly chose Saul as the choicest man at that moment, so that there was none like him among the children of Israel.
Tertullian (c. 160-c. 220), The Five Books against Marcion, Book II (trans. Peter Holmes), in Ante-Nicene Fathers, vol. 3 (ed. Alexander Roberts and James Donaldson; Buffalo: Christian Literature Publishing Co., 1885) — Against Marcion, Book II, ch. 24 ('Instances of God's Repentance...'), on 1 Samuel 15:11 and 35
Catholic
Aquinas holds that God's will is entirely unchangeable. Changing one's will differs from willing that certain things change: one may will something now and its contrary afterward, yet the will remains permanently the same. The will would change only if one began to will what was not willed before, or stopped willing what was willed before.
Thomas Aquinas (1225-1274), Summa Theologiae, Second and Revised Edition (trans. the Fathers of the English Dominican Province; New Advent online) — I, q. 19, art. 7 ('Whether the will of God is changeable?')
Orthodox
Scripture uses a show of anger to restrain the unbridled by fear. To those needing repentance, it says the Lord repents with them of the evil, and by God's repentance concerning Saul it warns the prosperous that their good fortune is no certain possession, though it seem to come from God.
Gregory of Nyssa (c. 335-c. 395), Answer to Eunomius' Second Book (trans. William Moore and Henry Austin Wilson), in Nicene and Post-Nicene Fathers, ser. 2, vol. 5 (New York: Christian Literature Publishing Co., 1893) — Answer to Eunomius' Second Book, the passage on Scripture's anthropopathic language (listing 'He repented that He had anointed Saul king')
Reformed / Protestant
Henry says repentance in God is not a change of his mind, as in us, but a change of his method or dispensation. He does not alter his will, but wills an alteration. The change was in Saul, who turned back from following God; God put this construction on the partiality of his obedience and the prevalence of his covetousness.
Matthew Henry (1662-1714), Commentary on the Whole Bible, vol. 2: Joshua to Esther (1706-1721) — Exposition of 1 Samuel 15:11 ('It repenteth me that I have set up Saul')
Evangelical
Wesley says repentance implies grief of heart and change of counsels, so it cannot be in God. But it is ascribed to God when he alters his method of dealing and treats a person as if he repented of the kindness he had shown.
John Wesley (1703-1791), Explanatory Notes upon the Old Testament (1765; CCEL's 'Wesley's Notes on the Bible') — Note on 1 Samuel 15:11