Short answer
The Bible says both. In 1 Samuel 15, God (﷾) says He repents (is sorry) that He made Saul king. A few verses later, Samuel says God (﷾) "is not a man, that he should repent". Then the chapter ends by saying the Lord repented that He had made Saul king. In Genesis 6, God (﷾) regrets that He made man at all, and it grieves Him at His heart. Yet Numbers 23:19 says God (﷾) does not repent. The Qur'an tells the story of man another way. When God (﷾) said He would place man on earth, the angels feared bloodshed. God (﷾) answered that He knew what they did not know. He knew everything before He made man, so He never regrets making him.
Full explanation
One chapter, two answers
In 1 Samuel 15 God (﷾) says He repents of making Saul king. Samuel then says God (﷾) is not a man that He should repent. The chapter closes by saying the Lord repented of making Saul king. The chapter says God (﷾) does it and says God (﷾) does not.
“It repenteth me that I have set up Saul to be king: for he is turned back from following me, and hath not performed my commandments. And it grieved Samuel; and he cried unto the Lord all night.”
“And also the Strength of Israel will not lie nor repent: for he is not a man, that he should repent.”
“And Samuel came no more to see Saul until the day of his death: nevertheless Samuel mourned for Saul: and the Lord repented that he had made Saul king over Israel.”
God's (﷾) regret over making man
Genesis goes further. It says God (﷾) was sorry that He had made mankind, and that it grieved Him at His heart. It puts the regret in God's (﷾) own mouth.
“And it repented the Lord that he had made man on the earth, and it grieved him at his heart.”
“And the Lord said, I will destroy man whom I have created from the face of the earth; both man, and beast, and the creeping thing, and the fowls of the air; for it repenteth me that I have made them.”
The Torah says God (﷾) does not repent
Numbers 23:19 says God (﷾) is not "the son of man, that he should repent". Yet Genesis 6 and 1 Samuel 15:35 say that He did.
“God is not a man, that he should lie; neither the son of man, that he should repent: hath he said, and shall he not do it? or hath he spoken, and shall he not make it good?”
“And it repented the Lord that he had made man on the earth, and it grieved him at his heart.”
“And Samuel came no more to see Saul until the day of his death: nevertheless Samuel mourned for Saul: and the Lord repented that he had made Saul king over Israel.”
How Christians read these verses
Christians hold that God's (﷾) nature never changes. They point to Malachi and James. They read "the Lord repented" as a way of describing how God (﷾) changes His dealings when people change, as Jeremiah 18 puts it.
“For I am the Lord, I change not; therefore ye sons of Jacob are not consumed.”
“Every good gift and every perfect gift is from above, and cometh down from the Father of lights, with whom is no variableness, neither shadow of turning.”
“If that nation, against whom I have pronounced, turn from their evil, I will repent of the evil that I thought to do unto them.”
Regret, then the Flood, then the cross Christians speak of
Follow the Bible's own account in order. In Genesis 6, God (﷾) saw that "every imagination of the thoughts of his heart was only evil continually". Then "it repented the Lord that he had made man on the earth", and He said, "I will destroy man whom I have created". The Flood followed, and only Noah's family was saved. After the Flood, God (﷾) says in Genesis 8:21 that He will not again curse the ground "for man's sake". He names man's heart again: "the imagination of man's heart is evil from his youth". The same evil heart appears before the Flood and after it. Then John 3:16 says God (﷾) "so loved the world, that he gave his only begotten Son". Christians understand this as the sacrifice on the cross. They say the cross was planned "before the foundation of the world" (1 Peter 1:20). If so, God (﷾) knew all that would follow before He made man. Yet Genesis 6:6 has Him regret making man, grieved at His heart. A plan fixed before creation and regret at having created do not fit together. In the Qur'an Allah knew before He made people what they would do, and never regrets His creation.
“And God saw that the wickedness of man was great in the earth, and that every imagination of the thoughts of his heart was only evil continually.”
“And it repented the Lord that he had made man on the earth, and it grieved him at his heart.”
“And the Lord said, I will destroy man whom I have created from the face of the earth; both man, and beast, and the creeping thing, and the fowls of the air; for it repenteth me that I have made them.”
“And all flesh died that moved upon the earth, both of fowl, and of cattle, and of beast, and of every creeping thing that creepeth upon the earth, and every man: All in whose nostrils was the breath of life, of all that was in the dry land, died. And every living substance was destroyed which was upon the face of the ground, both man, and cattle, and the creeping things, and the fowl of the heaven; and they were destroyed from the earth: and Noah only remained alive, and they that were with him in the ark.”
“And the Lord smelled a sweet savour; and the Lord said in his heart, I will not again curse the ground any more for man's sake; for the imagination of man's heart is evil from his youth; neither will I again smite any more every thing living, as I have done.”
“For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.”
“Who verily was foreordained before the foundation of the world, but was manifest in these last times for you,”
“Behold, thy Lord said to the angels: "I will create a vicegerent on earth." They said: "Wilt Thou place therein one who will make mischief therein and shed blood?- whilst we do celebrate Thy praises and glorify Thy holy (name)?" He said: "I know what ye know not."”
In the Qur'an, God (﷾) knew before He made man
Allah (﷾) announced that He would place man on the earth. The angels asked whether He would place there one who would spread corruption and shed blood. His answer was that He knows what they do not. The Qur'an adds that nothing, down to a falling leaf, is outside His knowledge. A God who knows the end before the beginning has nothing to regret.
“Behold, thy Lord said to the angels: "I will create a vicegerent on earth." They said: "Wilt Thou place therein one who will make mischief therein and shed blood?- whilst we do celebrate Thy praises and glorify Thy holy (name)?" He said: "I know what ye know not."”
“With Him are the keys of the unseen, the treasures that none knoweth but He. He knoweth whatever there is on the earth and in the sea. Not a leaf doth fall but with His knowledge: there is not a grain in the darkness (or depths) of the earth, nor anything fresh or dry (green or withered), but is (inscribed) in a record clear (to those who can read).”
Verse by verse
Jeremiah 18:8 — Nevi'im, Yirmeyahu, chapter 18, verse 8
“If that nation, against whom I have pronounced, turn from their evil, I will repent of the evil that I thought to do unto them.”
Cited for
When the Bible says God repented, it means God changed His dealings because people changed, as Jeremiah 18 explains. His nature stays the same.
The response
Jeremiah 18 speaks of a threat or a promise withdrawn when a nation changes. 1 Samuel 15:11 and 15:35 go further. God (﷾) is sorry for His own past act, "that I have set up Saul to be king". Genesis 6:6 is regret over His own act of creating man. A God who knew from the start how Saul and mankind would turn out has no reason to regret choosing or making them.
Where it lands — The conditional reading explains a changed decision, not sorrow over God's (﷾) own past choices, which is what 1 Samuel 15 and Genesis 6 describe.
Genesis 6:6 — Torah, Bereshit, chapter 6, verse 6
“And it repented the Lord that he had made man on the earth, and it grieved him at his heart.”
Cited for
"Repent" is language accommodated to human understanding. It describes how God's actions looked from the human side, not a real change in God.
The response
Genesis 6:6 is not a report of what people saw. It is the narrator telling what God (﷾) felt: "it grieved him at his heart". Genesis 6:7 puts "it repenteth me" in God's (﷾) own mouth. In 1 Samuel 15:11 the Lord Himself is sorry for His own past act, "that I have set up Saul to be king". That is sorrow over a decision He made, which 15:29 denies of Him. A figure of speech does not explain why the Bible gives God (﷾) regret over His own acts.
Where it lands — The Bible gives God (﷾) real regret over His own acts, and also denies that He repents.
1 Samuel 15:29 — Nevi'im, Shmuel Aleph, chapter 15, verse 29
“And also the Strength of Israel will not lie nor repent: for he is not a man, that he should repent.”
Cited for
Verse 29 means God will not revoke His sentence on Saul; verses 11 and 35 speak of His sorrow.
The response
In 1 Samuel 15:11 the Lord Himself is sorry for His own past act, "that I have set up Saul to be king". Genesis 6:6 says "it grieved him at his heart". That is sorrow over a decision He made, which 15:29 denies of Him. A figure of speech does not explain why the Bible gives God (﷾) regret over His own acts.
Where it lands — The chapter says God (﷾) is sorry for His own act, and also says He is not a man that He should repent.
Qur'an 2:106 — Surat al-Baqara (The Cow), verse 106
“None of Our revelations do We abrogate or cause to be forgotten, but We substitute something better or similar: Knowest thou not that Allah Hath power over all things?”
Cited for
The Qur'an also has God replace His rulings with new ones, so He changes His mind too.
The response
Replacing one ruling with another at the time God (﷾) planned is a teacher moving a class on, not regret. When the Qur'an replaces one revelation with another, it says Allah (﷾) knows best what He reveals (16:101). Islam also agrees that God (﷾) may withdraw a punishment when people turn back (10:98). That is mercy, not sorrow over what He has done. Nowhere does the Qur'an say God (﷾) was sorry for what He had done, as Genesis 6:6 and 1 Samuel 15:11 say.
Where it lands — A planned change of law is not repentance over one's own past act.
Qur'an 3:140 — Surat Al Imran (The Family of Imran), verse 140
“If a wound hath touched you, be sure a similar wound hath touched the others. Such days (of varying fortunes) We give to men and men by turns: that Allah may know those that believe, and that He may take to Himself from your ranks Martyr-witnesses (to Truth). And Allah loveth not those that do wrong.”
Cited for
The Qur'an says "that Allah may know those that believe" (3:140), so its God learns too.
The response
Muslims read this as the trial showing in the open what Allah (﷾) already knew. The Qur'an says every misfortune is recorded in a decree before it happens (57:22), and that He knew before He made man (2:30). Nowhere does it say God (﷾) was sorry for what He had done.
Where it lands — A trial shows what God (﷾) already knew. It is neither learning nor regret.
Perspectives
Mainstream Christian
God is unchanging. When the Bible says He repented, it describes His changed dealings with people who have changed, told in human terms.
The Bible teaches that God does not change: "I am the Lord, I change not."
Rebuttal
Islam agrees that God (﷾) does not change, as Malachi and James say. But the same Bible says the Lord repented twice in 1 Samuel 15, and that it grieved Him at His heart in Genesis 6:6. Numbers 23:19 says He is not a son of man, that He should repent, so the Bible gives two pictures of God (﷾).
God's "repenting" is His response when a nation turns from evil, as Jeremiah 18 explains.
Rebuttal
Islam agrees that God (﷾) may withdraw a threatened punishment when people turn back (10:98), as Jeremiah 18 says. But in 1 Samuel 15:11 God (﷾) is sorry for His own past act, "that I have set up Saul to be king". Genesis 6:6 is regret over making man. A God who knew from the start how Saul and mankind would turn out has nothing to regret.
Christians say that God grieves over sin but still loves people, and that the cross shows it.
Rebuttal
The issue is not sorrow at sin, but regret at making man: "it repenteth me that I have made them" (Genesis 6:7). Christians say the cross was planned "before the foundation of the world" (1 Peter 1:20). Then God (﷾) knew all this before He made man, and regret does not fit.
The Islamic perspective
God's (﷾) knowledge is complete, so He never regrets what He has done. A book that says He repents and says He does not, in the same chapter, cannot be His unchanged word.
1 Samuel 15 says God (﷾) repented of making Saul king (verses 11 and 35). The same chapter says He does not repent (verse 29).
Genesis 6 says God (﷾) regretted making mankind and was grieved at His heart.
Numbers 23:19 says God (﷾) is not a son of man, that He should repent.
The Qur'an teaches that Allah (﷾) knew, before He created man, all that the angels feared. Nothing is outside His knowledge.