Although the N.T. often speaks of Jesus as the Son of God, what's significant about the way the Apostle John uses that title?

It is useful to enumerate the uses of 'the Son of God' in John's Gospel. I am not including here the word 'Son' applied to Jesus, as these are ambiguous - it could mean Son of Man or Son of God. There are 15 such uses of 'the Son of God' or similar phrases in the Gospel of John. It is worth simply reading through all the instances to get a sense for how it is used in the Gospel of John, and avoid cherry-picking issues, and they are appended to the end of this answer.

1. Jesus is using an appropriate title - it is not blasphemy.

Let's look first at Jesus' words himself, at John 10:35-36.

"If he called them gods to whom the word of God came — and the Scripture cannot be broken — 36 then what about the One whom the Father sanctified and sent into the world? How then can you accuse Me of blasphemy for stating that I am the Son of God?"

Jesus is quoting Psalm 82, of which most Jews would have been familiar. Psalm 82:6 doesn't just call the judges 'gods', but immediately afterward calls them the equivalent of 'sons of God'.

"I have said, ‘You are gods; you are all sons of the Most High.’

Note also what Jesus says at 10:35-36, that each judge is called theos (collectively theoi or 'gods') because the word of God came to them. Did the word of God come to Jesus? Of course - that's a big theme in the Gospel of John. So Jesus is properly 'theos' according to the scriptures the Jews give authority to - but he is only claiming to be 'son of theos' here, a lesser title, so certainly not blasphemy because not only has the word of God come to him, but He has been uniquely sent by God.

Note what is explicit here in Jesus' statement - calling someone 'theos' doesn't mean that someone is God. It's just a manner of speaking, because they reflect the word (or message) of God. So does 'Son of God' mean Jesus is God? No, not for John. Rather, Jesus has not only been sent the word or message of God but also is uniquely sent into the world on a mission from God - you could say he has been uniquely chosen, or 'anointed', i.e., the Christ.

2. The title is linked to being King.

Next, let's move to Nathanael's use of 'the Son of God', which is early in John's Gospel, at 1:49.

What is Nathanael thinking of here? Note he combines 'King of Israel' with 'the Son of God'. Were there any other kings of Israel who were considered sons of God? There are two obvious figures, David and Solomon.

Psalm 89:26-7, describing David, says

"He will call to Me, ‘You are my Father, my God, the Rock of my salvation.’ 27 I will indeed appoint him as My firstborn, the highest of the kings of the earth."

2 Samuel 7:14, arguably describing Solomon, says

"I will be his Father, and he will be My son."

The Son of God will be a King, like David or Solomon.

3. Although 'Son of God' has been used before for David and arguably Solomon, Jesus is unique.

This has already been indicated with what Jesus says at 10:35-36, looked at above. But this idea is repeated multiple times in the Gospel of John. The Son of God = a unique savior figure, greater than any king or patriarch who has come before (Abraham, Moses, David, Solomon).

John 1:14 'the one and only Son'. John 1:18 'the one and only Son'. John 3:16 'his one and only Son'. John 3:18 'God's one and only Son'. The entire tenor of Jesus' immediate response at John 5:18-26.

4. There is a strong conceptual connection between 'the Son of God' and 'the Christ'.

John 1:34 has this with textual variants of what John the Baptist says. One has him saying "the Son of God", another "the Christ (chosen one)".

This linkage is further reinforced by John the Baptist in John 3 denying he himself is the Christ - the obvious inference is that Jesus instead is the Christ.

Similarly, John 6:69 has "We believe and know that You are the the Christ, the Son of the living God."

Finally, Martha says at John 11:27 "You are the Christ, the Son of God, who was to come into the world."

For the author John, the Son of God is a co-referent for the uniquely chosen one (Christ) of God.

5. So to believe in Jesus as the Son of God is to believe that Jesus is the uniquely sent, salvific King - the Christ (chosen one) of OT prophecies.

Consider Martha's response to Jesus at John 11:25-27.

"Jesus said to her, “I am the resurrection and the life. Whoever believes in Me will live, even though he dies. 26 And everyone who lives and believes in Me will never die. Do you believe this?” 27 “Yes, Lord,” she answered, “I believe that You are the Christ, the Son of God, who was to come into the world.”"

This is a reference to prophesies of the coming Christ, or Messiah, or King of Israel, or Son of God. The titles are co-referents, and used in combination to pick out the unique figure (such as is referred to in Ps. 2).

To claim to be such a figure if it isn't true is of course blasphemy. This is what the Jews are alleging at John 19:7, and what Jesus is charged with after He affirms He is indeed 'the Christ, the Son of God', in the trial before the Sanhedrin in the synoptic Gospels.

Appendix

Below are all instances of the use of the phrase 'the Son of God' (or similar phrases) in John's Gospel.

John 1:14

"We have seen His glory, the glory of the one and only Son [or the only begotten or the unique one] from the Father, full of grace and truth."

John 1:18

"No one has ever seen God, but the one and only Son, who [is Himself God and?] is at the Father’s side, has made Him known."

John 1:34

"The man on whom you see the Spirit descend and rest is He who will baptize with the Holy Spirit.’ 34 I have seen and testified that this is the Son [or the chosen one, i.e., Christ] of God."

John 1:49

"“Rabbi,” Nathanael answered, “You are the Son of God! You are the King of Israel!”"

John 3:16-18

"For God so loved the world that He gave His one and only Son, that everyone who believes in Him shall not perish but have eternal life. 17 For God did not send His Son into the world to condemn the world, but to save the world through Him. 18 Whoever believes in Him is not condemned, but whoever does not believe has already been condemned, because he has not believed in the name of God’s one and only Son."

John 5:18-26

"Because of this, the Jews tried all the harder to kill Him. Not only was He breaking the Sabbath, but He was even calling God His own Father, making Himself equal with God. 19 So Jesus replied, “Truly, truly, I tell you, the Son can do nothing by Himself, unless He sees the Father doing it. For whatever the Father does, the Son also does. 20 The Father loves the Son and shows Him all He does. And to your amazement, He will show Him even greater works than these. 21 For just as the Father raises the dead and gives them life, so also the Son gives life to whom He wishes. 22 Furthermore, the Father judges no one, but has assigned all judgment to the Son, 23 so that all may honor the Son just as they honor the Father. Whoever does not honor the Son does not honor the Father who sent Him. 24 Truly, truly, I tell you, whoever hears My word and believes Him who sent Me has eternal life and will not come under judgment. Indeed, he has crossed over from death to life. 25 Truly, truly, I tell you, the hour is coming and has now come when the dead will hear the voice of the Son of God, and those who hear will live. 26 For as the Father has life in Himself, so also He has granted the Son to have life in Himself."

John 6:69

"We believe and know that You are the the Christ, the Son of the living God [or Holy One of God]."

John 9:35

"When Jesus heard that they had thrown him out, He found the man and said, “Do you believe in the Son of God [or Son of Man]?"

John 10:35-36

"If he called them gods to whom the word of God came—and the Scripture cannot be broken— 36 then what about the One whom the Father sanctified and sent into the world? How then can you accuse Me of blasphemy for stating that I am the Son of God?"

John 11:4

"When Jesus heard this, He said, “This sickness will not end in death. No, it is for the glory of God, so that the Son of God may be glorified through it.”"

John 11:27

“Yes, Lord,” she answered, “I believe that You are the Christ, the Son of God, who was to come into the world.”

John 19:7

"“We have a law,” answered the Jews, “and according to that law He must die, because He declared Himself to be the Son of God.”"

John 20:31

"But these are written so that you may believe that Jesus is the Christ, the Son of God, and that by believing you may have life in His name."

Q: What are the ‘signs’ John gives in his gospel that point to Jesus as the Christ, the Son of God?

A: There are seven miraculous signs in John’s gospel that suggest Jesus was much more than the ‘son of man’, a title Jesus used of himself during his earthly ministry. [1] For example, after the first sign, John writes:

He thus revealed his glory and his disciples put their faith in him (John 2:11).

Here are the seven best known miraculous signs that point to the deity of Jesus:

Wedding miracle at Cana

Healing the official's son

Healing the paralytic

Feeding the 5000

Walking on water

Healing the blind man

Raising Lazarus from the dead

Q: What is significant about the way John uses the title ‘Son of God’?

A: John used the title ‘Son of God’ to show the divinity of Jesus and his relationship with his Father [2]

Jesus did many other miraculous signs in the presence of his disciples, which are not recorded in this book. But these are written that you may believe that Jesus is the Christ, the Son of God, and that by believing you may have life in his name (John 20:30-31).

John likely wrote his gospel account after A.D. 85 and his view of Christ was of the risen Christ, in heavenly glory, yet to return to earth. Clement of Alexandria stated that John wrote to supplement the accounts in the other gospels. (Clement, Ecclesiastical History, 6.14.7)

By the time the aged apostle finished writing, his view of Christ was focused on the glorious, risen Christ in heaven. Before John died, the doctrine of the deity of Christ was well established in the Acts of the Apostles and Paul’s letters to the churches. John’s emphasis was on Christ as the glorified Son of God, who would return to gather his own.

Conclusion: John selected seven miraculous signs given during the ministry of the Son of Man on earth to signify what was to be revealed after the resurrection and ascension of Jesus when he entered into his glory. John’s deliberate use of the title ‘Son of God’ is to reveal the divine relationship between Father and Son so that those who believe will be brought into union with them, adopted into God’s family as heirs to the promise.

Additional relevant information found during the course of my research:

When Jesus was in the wilderness, the tempter taunted Jesus by saying if he was the Son of God he could turn the stones into bread (Matthew 4:3). Even God’s arch-enemy knew who Jesus really was!

By divine revelation, Peter realised that Jesus was no mere mortal but was the Christ, the Son of God. See Matthew 16:16, Mark 8:29, and Luke 9:20.

John the Baptist testified that Jesus is the Son of God (John 1:34).

Nathanael declared that Jesus is the Son of God, the King of Israel (John 1:49).

[1] Jesus used the title "Son of Man" to describe Himself and his humanity while on earth: https://www.biblegateway.com/quicksearch/?qs_version=NKJV&quicksearch=Son+of+Man&begin=47&end=73 (New King James Version)

[2] John used the title ‘Son of God’ to show the divinity of Jesus and his relationship with his Father: https://www.biblegateway.com/quicksearch/?qs_version=NKJV&quicksearch=Son+of+God&begin=50&end=50 (New King James Version)

The Gospel of John is much more focused on the aspect of relationship. Who each one is in relationship to others is important to John's message. Especially is this true with respect to Jesus.

Here are several examples of this, none of which are found in the other gospels:

The Father loveth the Son, and hath given all things into his hand. (John 3:35, KJV)

For the Father loveth the Son, and sheweth him all things that himself doeth: and he will shew him greater works than these, that ye may marvel. (John 5:20, KJV)

Ye have heard how I said unto you, I go away, and come again unto you. If ye loved me, ye would rejoice, because I said, I go unto the Father: for my Father is greater than I. (John 14:28, KJV)

And, of course, the most famous text in the Bible:

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. (John 3:16, KJV)

Repeatedly, John focuses on the beloved Son of God. The Father-Son relationship is the means of demonstrating God's love toward us. And God, who is said to be the Father, is also called our Father.

Jesus shares with us his Father--his God.

Jesus saith unto her, Touch me not; for I am not yet ascended to my Father: but go to my brethren, and say unto them, I ascend unto my Father, and your Father; and to my God, and your God. (John 20:17, KJV)

To John, Jesus' life is about restoring us to sonship with our Heavenly Father. This theme extends from the first chapter to the last in his book.

But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name: (John 1:12, KJV)

Conclusion

Sonship is a central theme of John's message. Jesus, as the only begotten Son of God, desires for us to become his brothers, and joint heirs with him as sons of God. Jesus even lovingly calls his disciples his "brethren."

Jesus saith unto her, Touch me not; for I am not yet ascended to my Father: but go to my brethren, and say unto them, I ascend unto my Father, and your Father; and to my God, and your God. (John 20:17, KJV)