The opposing Jews wanted to stone Jesus for claiming that he had “seen Abraham,” although, as they said, Jesus was “not yet 50 years old.” (John 8:57)
The following quotes are taken from the NWT study notes on John 8:58:
Jesus’ response was to tell them about his pre-human existence as a
mighty spirit creature in heaven before Abraham was born. The argument
that the Greek expression used here, ἐγὼ εἰμί / ego eimi (rendered “I
am” in most Bible translations), is an allusion to the Septuagint
rendering of Exodus 3:14 and that both verses should be rendered the
same way. Exodus 3:14 declares God's name, for Moses asked what's the
name of God. In the context for John 8:58, however, the action
expressed by the Greek verb eimiʹ started “before Abraham came into
existence” and was still in progress.
It is therefore properly translated “I have been” rather than “I am,”
and several ancient and modern translations use wording similar to “I
have been.”
Imagine God would have said something slightly different to Moses. Imagine he would simply have said "Yahweh". Do you suggest Jesus would have said to the Jews "Before Abraham was, Yahweh"? That wouldn't make any sense.
Also, if ego eimi was such a trigger term, then why did the Jews act completely differently, when a few verses later, the blind man who got cured by Jesus made himself known to them by saying exactly the same words : ego eimi (I am he) in John 9:9? Did the Jews pick up stones to kill this man, whom they knew to be previously blind, for this claim?
At John 14:9, the same form of the Greek verb ei·miʹ is
used to render Jesus’ words: “Even after I HAVE BEEN with you men for
such a long time, Philip, have you not come to know me?” Most
translations use a similar wording, showing that depending on the
context there is no valid grammatical objection to rendering ei·miʹ as
“have been.” (Other examples of rendering a present tense Greek verb
using a present perfect tense verb are found at Luke 2:48; 13:7;
15:29; John 15:27; Acts 15:21; 2 Corinthians 12:19; 1 John 3:8.)
Also, Jesus’ reasoning recorded in John 8:54, 55 shows that he was not trying to portray himself as the same ontological being as his Father. On the contrary, in John 8:17,18 Jesus refers to Jewish law: "In your Law it is written that the testimony of two people is true. I am the one who bears witness about myself, and the Father who sent me bears witness about me.” In John 5:31 Jesus said: "If I alone bear witness about myself, my testimony is not true." Jesus thus made it very clear that there is an ontological difference between himself and the Father. Otherwise, the testimony about Jesus being the Son of God would not have been acceptable evidence according to Jewish Law (established by God I might add).
Possible reasons for the Jews to pick up stones, and attempt to kill Jesus
According to Deuteronomy 18:20-22, anyone who claims to be a prophet of God should be condemned to death if they proclaim something false in the name of God.
Both criteria fit the case at hand: Jesus directly and repeatedly claimed to speak in the name of God. Let's look at the following examples (all taken from John’s gospel):
7:16: My doctrine is not mine, but his that sent me.
7:28: I have not come of Myself, but He who sent Me is true
8:16: the Father who sent me.
8:18: the Father who sent me bears witness about me.
8:26: He who sent me is true
8:28: I do nothing of Myself; but as My Father taught Me, I speak these things
8:40: [I am] a Man who has told you the truth which I heard from God
8:42: I proceeded forth and came from God; nor have I come of Myself, but He sent Me.
This list makes it clear that Jesus openly claimed to be a prophet, someone who speaks in the name of God. And immediately before Jesus uttered the words of verse 58, Jesus was accused of the fact that Abraham could never have seen him, because Jesus was not yet 50 years old, while Abraham had already been dead for many centuries.
Jesus' answer that he had already existed before Abraham's birth could be used as an excuse to say that Jesus had said something false in the name of God. For it seemed clear to everyone that Jesus could not have been that old. And thus, according to his opponents, he had spoken a word in the name of God that was false, thus fulfilling the characteristic of a false prophet from Deuteronomy 18. This did not require Jesus to make a "claim to divinity".
But there are even more reasons to stone Jesus. Anyone reading the words from John 8:48-52 today is inclined to misunderstand them. There Jesus is accused of being possessed by a demon and of being a Samaritan. While today we associate "Samaritan" with the parable of the Good Samaritan, the accusation of being “possessed by a demon” sounds like "you’re a lunatic”. Both are misleading.
According to John 8:20, the discussion took place in the treasury of the temple, which in turn was located in the so-called court of the women. According to Exodus 15:31, a person who was ritually unclean and came into the temple (apart from the court of the Gentiles) was punished with death.
Although there was no direct commandment that a possessed person was unclean, this was a possible excuse that the presence of a possessed person desecrated the temple.
Furthermore, possession can be attributed to someone being a spiritualist, this connection is obvious. And according to Exodus 20:27 and Deuteronomy 18:10-12, this was again a crime punishable by death. The accusation: "You have a demon!" was therefore also an attempt to find an excuse to kill Jesus.
It was similar with the accusation: "You are a Samaritan!" This was not an accusation that accused Jesus of mercy. What we find in the "Jewish Antiquities" section XVIII.2.2 of Flavius Josephus is interesting. According to the description there, Samaritans crept into the temple about twenty years before Jesus' appearance and deliberately defiled it with parts of corpses. From the account, we can conclude that Samaritans were obviously not allowed in the temple (otherwise they would not have had to sneak in) and, moreover, they were known from that time onwards for trying to desecrate the temple.
It was also common knowledge that Jesus was a Jew from Galilee. Anyone who accused him of being a Samaritan automatically accused him of apostasy from the Jewish faith. This was also a pretext that was enough to stone someone who came to the temple. Decades later, when a lynch mob tried to kill Paul in the temple, one of the accusations according to Acts 21:28 was also apostasy; and the accusation there also ended with an attempted murder of Paul.
The above is a possibility and I cannot claim that it must have been so with 100% certainty. However, in the context of Jesus' claim to speak as God's representative, these accusations result in another obvious attempt to portray Jesus as a false prophet.
It is clear that a prophet of God is guided by the Holy Spirit. This rules out the possibility of the same person being possessed by a demon at the same time. So anyone who accused Jesus of being possessed was also accusing him of claiming to speak in the name of God, since a possessed person cannot be God's prophet.
And of course a prophet had to be an Israelite. The Bible contains no examples of Jehovah sending a message to his people through a non-Israelite. When Jesus is referred to as a Samaritan, it is again denied that he could be a true prophet. As in chapter 7:52, the Jewish elite even denied that a prophet could come from Galilee (which is obviously wrong, since according to 2 Kings 14:25 Jonah came from Gath-Hepher in Galilee), although they only said this out loud "in the inner circle". Obviously, in their opinion, a prophet could much less come from Samaria. So if a Samaritan claimed to be sent by God, then in the opinion of the Jewish elite he must obviously be a false prophet.
Therefore, both accusations definitely served to portray Jesus as a false prophet; and that was a reason to stone him.
Did the Pharisees even understand Jesus?
Furthermore, reread the whole account starting in John 8:12 - Did the Pharisees understand Jesus in any way or grasp what he was saying? They repeatedly misunderstood Jesus - in fact, there was not a single thing they got correct, as is clear from the below list of questions/comments they uttered (all from John's gospel again):
8:19: Then they said to him, "Where is your Father?
Obviously they had not understood that Jesus was talking about God, otherwise they would hardly have asked the question in this way.
8:22: Then the Jews said, "Surely He does not want to kill Himself by saying, 'Where I am going, you cannot come?
Here, too, they had not understood what Jesus was talking about; the assumption of suicide was possibly wishful thinking, but had nothing to do with Jesus' intentions.
8:25: Then they said to him, "Who are you?
Jesus had already pointed out several times that he was a prophet, but they did not want to/could not understand this.
8:27: They did not recognise that he was speaking to them about the Father.
An unmistakable statement
8:33: They answered him, "We are Abraham's descendants and have never been anyone's slaves. How do you say: You shall be free?
They do not understand what kind of freedom Jesus is talking about.
8:39: They answered and said to him, "Abraham is our father.
8:41: They said to him, "We were not born through fornication; we have one Father, God.
It took them three attempts to understand what Jesus was accusing them of. Incidentally, this is the only point where we can say with some certainty that they understood Jesus (in the end).
8:48: The Jews answered and said to him, "Are we not right in saying that you are a Samaritan and have a demon?
8:52: The Jews therefore said to him, "Now we recognise that you have a demon.
These accusations are obviously false and are based on the fact that they (want to) misunderstand Jesus.
8:53: What are you making of yourself?
This question shows that nothing Jesus has said up to this point has been understood.
8:57: Then the Jews said to him, "Are you not yet fifty years old and have you seen Abraham?
This question also stems from a misunderstanding. Jesus thus far had not claimed to have seen Abraham.
Jesus was even so frustrated with them that he said „Why do I speak to you at all?“ (John 8:25 - NRSV) When they asked him why he said he saw Abraham, as he wasn’t even 50 years old, he said „ego eimi“ - guess what might have happened? They misunderstood him again!
So the majority understanding of John 8:58 is that the Pharisees who did get Jesus wrong the whole time, all of a sudden had an epiphany and suddenly understood him correctly by him claiming to be God, only by him saying „ego eimi“ - you recognize how absurd that argument appears to be? It shows me what is often true - the majority isn't always right!
Going even further into the whole context of John 8, you'll see that they (the Pharisees) continuously felt their authority being questioned, and they became more and more exposed by Jesus for their false reasonings. It could have been this, that caused the Pharisees to lose it and attempt to stone Jesus, as by stating that Jesus had seen Abraham, because he was older than him, he claimed even higher authority than Abraham.
Seniority was a very important aspect of granting authority in Jewish culture (Job 12:12). For the Pharisees, Abraham was the highest human authority, and they understood Jesus to claim higher authority than themselves, which was unacceptable insubordination - they hated him for it! This happened on numerous occasions, and they wanted to find an excuse to kill Jesus (Mark 11:18; Mark 14:1; John 7:30,32,44,45; Luke 19:47; 20:19)
So, it wasn't their fear of God that caused them to perceive Jesus' word as sinful! It was their selfish pride that caused them to act in utter outrage because Jesus had dared to question their authority repeatedly and established that his authority was higher than theirs.