Short answer
John says the crowd tried to stone him because they heard him as making himself God. That was the crowd's charge, not his claim. In John 10 he answered it. He said he had called himself the Son of God. He pointed to their own scripture, which calls those who received God's (﷾) word "gods". In John 5 the same charge came after he healed on the Sabbath and called God his Father. He replied: the Son can do nothing of himself. In John 8 he called himself "a man" who told the truth he heard from God (﷾). Being stoned was no sign of claiming to be God: the prophet Zechariah was stoned at a king's command for giving God's (﷾) message. Islam teaches that messengers often met rejection when they brought what people did not want. Jesus was one of them, a servant of God (﷾) who never claimed to be God.
Full explanation
The scenes Christians point to
John tells of two scenes in Jerusalem. After Jesus says "Before Abraham was, I am", the crowd picks up stones. After he says he and the Father are one, they say he makes himself God. Christians read the crowd's anger as proof that he claimed to be God. The people in these scenes are a crowd in Jerusalem. Jesus and his followers were Jews too.
“Jesus said unto them, Verily, verily, I say unto you, Before Abraham was, I am.”
“Then took they up stones to cast at him: but Jesus hid himself, and went out of the temple, going through the midst of them, and so passed by.”
“I and my Father are one.”
“The Jews answered him, saying, For a good work we stone thee not; but for blasphemy; and because that thou, being a man, makest thyself God.”
How Jesus answered the charge
When the crowd said he made himself God, Jesus did not answer that he was God. He said his claim was to be the Son of God. He pointed to their own scripture, the Psalms. There, those to whom the word of God (﷾) came are called gods and children of the Most High. A title that scripture gives to men cannot prove that the one who bears it is God.
“Jesus answered them, Is it not written in your law, I said, Ye are gods? If he called them gods, unto whom the word of God came, and the scripture cannot be broken; Say ye of him, whom the Father hath sanctified, and sent into the world, Thou blasphemest; because I said, I am the Son of God?”
“I have said, Ye are gods; and all of you are children of the most High.”
The reason John gives in chapter 5
John gives Sabbath healing (healing on the Jewish day of rest) as a reason they sought to kill him. He says they wanted him dead because he "not only had broken the sabbath" but also made himself equal with God (﷾). Yet Jesus said he had not come to destroy the law, not even its smallest letter. He also said it is lawful to do good on the Sabbath. So the Sabbath charge is one his own words deny. To the charge of making himself equal with God (﷾), Jesus answers at once: the Son can do nothing of himself.
“And therefore did the Jews persecute Jesus, and sought to slay him, because he had done these things on the sabbath day.”
“Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil. For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled.”
“How much then is a man better than a sheep? Wherefore it is lawful to do well on the sabbath days.”
“Therefore the Jews sought the more to kill him, because he not only had broken the sabbath, but said also that God was his Father, making himself equal with God.”
“Then answered Jesus and said unto them, Verily, verily, I say unto you, The Son can do nothing of himself, but what he seeth the Father do: for what things soever he doeth, these also doeth the Son likewise.”
What Jesus called himself
In the same talk as "before Abraham", eighteen verses earlier, Jesus tells the crowd what he is. He is a man who told them the truth he heard from God (﷾). In John 5 he says he can do nothing of himself and seeks the will of the one who sent him. In John 8:58, "I am" is the everyday Greek ego eimi. The Greek Old Testament, the old Greek translation of the Hebrew Bible, gives God's (﷾) name in Exodus 3:14 as ho on, "the One who is". Jesus does not say it. The page on that verse takes this further.
“But now ye seek to kill me, a man that hath told you the truth, which I have heard of God: this did not Abraham.”
“I can of mine own self do nothing: as I hear, I judge: and my judgment is just; because I seek not mine own will, but the will of the Father which hath sent me.”
In the Greek Old Testament God says to Moses "ego eimi ho on", "I am the One who is", and the name Moses is to give the people is "ho on", "the One who is": "ho on has sent me to you".
Stoning does not prove a claim to be God
In Matthew, Jesus describes Jerusalem as the city that kills the prophets and stones those sent to it. Jesus spoke as a Jew to his own city, as Israel's prophets had done. The Hebrew Bible records one such stoning. Zechariah, son of the priest Jehoiada, spoke God's (﷾) message to the people. He was stoned in the Temple court at the king's command. The text says nothing of any claim by him to be God.
“O Jerusalem, Jerusalem, thou that killest the prophets, and stonest them which are sent unto thee, how often would I have gathered thy children together, even as a hen gathereth her chickens under her wings, and ye would not!”
“And the Spirit of God came upon Zechariah the son of Jehoiada the priest, which stood above the people, and said unto them, Thus saith God, Why transgress ye the commandments of the Lord, that ye cannot prosper? because ye have forsaken the Lord, he hath also forsaken you. And they conspired against him, and stoned him with stones at the commandment of the king in the court of the house of the Lord.”
What the Qur'an says
The Qur'an says messengers have often met rejection when they brought a message people did not want. It names Jesus among the messengers sent to the Children of Israel. In 2:87 it says: "Is it that whenever there comes to you a messenger with what ye yourselves desire not, ye are puffed up with pride?- Some ye called impostors, and others ye slay!" The Qur'an also says that "no bearer of burdens can bear the burden of another" (6:164). No one today answers for what others did long ago.
“We gave Moses the Book and followed him up with a succession of messengers; We gave Jesus the son of Mary Clear (Signs) and strengthened him with the holy spirit. Is it that whenever there comes to you a messenger with what ye yourselves desire not, ye are puffed up with pride?- Some ye called impostors, and others ye slay!”
“Say: "Shall I seek for (my) Cherisher other than Allah, when He is the Cherisher of all things (that exist)? Every soul draws the meed of its acts on none but itself: no bearer of burdens can bear the burden of another. Your goal in the end is towards Allah: He will tell you the truth of the things wherein ye disputed."”
Verse by verse
John 10:33 — Gospel of John, chapter 10, verse 33
“The Jews answered him, saying, For a good work we stone thee not; but for blasphemy; and because that thou, being a man, makest thyself God.”
Cited for
Christians say the crowd heard "I and my Father are one" as a claim to be God, and Jesus did not correct them.
The response
He answered them in the very next words. He said they accused him of blasphemy "because I said, I am the Son of God". He pointed to the Psalm that calls those to whom the word of God (﷾) came "gods" and "children of the most High". When he adds "the Father is in me, and I in him", he prays the same for his disciples in John 17.
Where it lands — Jesus answered the charge by calling himself the Son of God, a title scripture gives to men, so the accusers' charge is not his claim.
John 5:18 — Gospel of John, chapter 5, verse 18
“Therefore the Jews sought the more to kill him, because he not only had broken the sabbath, but said also that God was his Father, making himself equal with God.”
Cited for
John says Jesus made himself equal with God, and that is why they sought to kill him.
The response
Verse 18 reports why they sought to kill him. Jesus answers in the next verse: "The Son can do nothing of himself." A few verses later he adds, "I can of mine own self do nothing." One who can do nothing of himself is not equal with God (﷾). The honour of 5:23 (that all should honour the Son) follows from the Father's having "committed all judgment unto the Son". That is a given authority, not his own.
Where it lands — Verse 18 gives the reason they sought to kill him. His reply begins by saying the Son can do nothing of himself, which is not how God (﷾) speaks of Himself.
Perspectives
Mainstream Christian
Christians hold that the crowd understood Jesus correctly. They say he claimed to be God, and that the crowd tried to stone him for blasphemy, meaning an insult to God.
In John 10:33 the crowd says that Jesus, being a man, makes himself God. Christians hold that they understood him rightly.
Rebuttal
John does record the charge, but the crowd speaks it. Jesus answers that he called himself the Son of God, and the Psalm calls those who received God's (﷾) word "gods". He also says the Father is in him, and he prays the same oneness for his disciples (John 17:21).
Christians read "Before Abraham was, I am" as an echo of God's name in Exodus 3:14.
Rebuttal
The stones show what the crowd thought, not what he meant. In the same talk he calls himself "a man that hath told you the truth". The Greek Exodus names God ho on, which Jesus does not say; the blind man says ego eimi too (John 9:9).
The charge in John
In John, the charge that Jesus made himself God is spoken by the men who were seeking his death. His own words in the same Gospel point the other way.
The charge that Jesus made himself God is spoken by those seeking to kill him. Jesus answers it by calling himself the Son of God, a title scripture gives to men.
In Matthew, Mark and Luke, the usual name Jesus uses for himself is "the Son of man".
The Islamic perspective
Jesus was a prophet sent by God (﷾). He answered the charge of making himself God by calling himself the Son of God, a title scripture gives to men. He also called himself a man sent with the truth from God (﷾).
Jesus spoke as a Jew to his own city, as Israel's prophets had done. He said Jerusalem stones those sent to it.
The Hebrew Bible records the stoning of the prophet Zechariah son of Jehoiada in the Temple court.
Answering the charge of blasphemy, Jesus said he had called himself the Son of God. He cited scripture that calls those to whom the word of God (﷾) came gods.
A talk in John ends with the stones of John 8:59. In it, Jesus calls himself a man who told the truth he heard from God (﷾).
John gives a Sabbath healing as the first reason they sought to kill him. Yet Jesus said he had not come to destroy the law.
The Qur'an teaches that messengers have often met rejection, and that some sent to the Children of Israel were called impostors and some were killed. It also teaches that no one bears another's burden.