Jesus never said he is the Son as God, but instead he said that he is the Son of God which means:

• the Son has is distinct from the Father
• the Son has a Father that approves him
• the Son has his Father's authority or permission

Translation

Some bible translations like the KJV state that Jesus is the "only begotten Son of the Father":

I will declare the decree: the LORD hath said unto me, Thou art my Son; this day have I begotten thee. -- Psalm 2:7 (KJV); Acts 13:33; Hebrews 1:5, 5:5

And the word was made flesh, and dwelt among us, (and we beheld his glory, the glory as of the only begotten of the Father) full of grace and truth. -- John 1:14 (KJV)

The only begotten son, who is in the bosom of the Father, He has declared him. -- John 1:18 (NKJV)

In this was manifested the love of God toward us, because that God sent his only begotten Son into the world, that we might live through him. -- 1 John 4:9 (KJV)

The Greek word translated as "only begotten" is:

The literal translation seems to mean that Jesus was the only Son that was ever born to God. However, in John 3 we read that God actually has many "reborned sons".

Reborned sons

There was a man of the Pharisees, named Nicodemus, a ruler of the Jews:

The same came to Jesus by night, and said unto him, "Rabbi, we know that thou art a teacher come from God: for no man can do these miracles that thou doest, except God be with him."
Jesus answered and said unto him, "Verily, verily, I say unto thee, Except a man be born again, he cannot see the kingdom of God."
Nicodemus saith unto him, "How can a man be born when he is old? can he enter the second time into his mother's womb, and be born?"
Jesus answered, "Verily, verily, I say unto thee, Except a man be born of water and of the spirit, he cannot enter into the kingdom of God.

• That which is born of the flesh is flesh; and
• that which is born of the spirit is spirit. Marvel not that I said unto thee, Ye must be born again. The wind bloweth where it listeth, and thou hearest the sound thereof, but canst not tell whence it cometh, and whither it goeth: so is every one that is born of the spirit."
  Nicodemus answered and said unto him, "How can these things be?"
  Jesus answered and said unto him, "Art thou a master of Israel, and knowest not these things? Verily, verily, I say unto thee, We speak that we do know, and testify that we have seen; and ye receive not our witness. If I have told you earthly things, and ye believe not, how shall ye believe, if I tell you of heavenly things?"

And no man hath ascended up to heaven, but he that came down from heaven, even the Son of man which is in heaven. And as Moses lifted up the serpent in the wilderness, even so must the Son of man be lifted up: That whosoever believeth in him should not perish, but have eternal life. 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. For God sent not His Son into the world to condemn the world; but that the world through him might be saved. He that believeth on him is not condemned: but he that believeth not is condemned already, because he hath not believed in the name of the only begotten Son of God.

-- John 3:1-18 (KJV)

Jesus was not referring to the physical birth of a person as he corrected Nicodemus. Instead, Jesus refers to the "birth of water" and "birth of the spirit" which modern Christians call "baptism".

Therefore, some would argue that "the only begotten Son of God" means "the only baptized Son of God", because the purpose of baptism is to reborn the spirit and Jesus himself was baptized.

The context here is that Jesus was dealing with condemnation and everlasting life. John did not record the details of the entire conversation but concluded that it is necessary that we should believe that God sent His "only begotten Son" to save us, so that we "should not perish, but have everlasting life". It was very important for John to highlight multiple times in his book that Jesus was "the only begotten Son of God".

The only Son of God

The word "only" which came from the Greek word "monos", means "alone". This means "the only begotten Son of God" had to be a unique attribute to Jesus which sets him apart from every other person.

• Jesus was not the only (alone) one who was baptized. John the Baptist had baptize many people before Jesus, so it does not make sense to say Jesus is "the only baptized Son of God".
• Jesus was not the only "son of God", because there were many other "sons of God" named in the Old Testament before Jesus was born.

Anyone can be adopted into "sonship" by another Father. If this was John's intention it would have been sufficient to write "Jesus was a Son of God" (like many other), but that is not what he wrote.

If one study the root of the Greek word "monogenés":

monogenḗs (from 3411 /misthōtós, "one-and-only" and 1085 /génos, "offspring, stock") -- HELP Word-studies

Which literally means Jesus was "the only offspring Son of God" where all other sons were "adopted" by God. This is why Jesus was considered "the Son of God" and not just a son of God. For example the same language is used in Hebrews:

By faith Abraham, when he was tried, offered up Isaac: and he that had received the promises offered up his only begotten son. -- Hebrews 11:17 (KJV)

Isaac was not Abraham's only biological son. At that time Abraham already had another elder son, called Ishmael (Genesis 16:11). However, Ismael was rejected by the family. In this context it meant Isaac was the only heir of Abraham. If this was the understanding of the Greek word "monogenḗs", then it also makes sense in Jesus context who were the greatest heir of all time.

This could also explain why it was such a revelation for many people to discover this unique position of Jesus, for example:

Identification

Disciples

John

And I have seen and have borne witness that this (Jesus) is the Son of God. -- John 1:39 (ESV)

Now Jesus did many other signs in the presence of the disciples, which are not written in this book; 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. -- John 20:30

Mark

The beginning of the gospel of Jesus Christ, the Son of God. -- Mark 1:1 (ESV)

Martha

She said to him (Jesus), “Yes, Lord; I believe that you are the Christ, the Son of God, who is coming into the world.” -- John 11:27 (ESV)

Paul

And immediately he (Paul) proclaimed Jesus in the synagogues, saying, “He is the Son of God.” -- Acts 9:20 (NKJV)

For the Son of God, Jesus Christ, whom we proclaimed among you... -- 2 Corinthians 1:19 (NKJV)

Peter

Simon Peter replied, “You are the Christ, the Son of the living God.” -- Matthew 16:16 (ESV)

Author of Hebrews

Since then we have a great high priest who has passed through the heavens, Jesus, the Son of God, let us hold fast our confession. -- Hebrews 4:14 (NKJV)

Non-disciples

Gabriel

And behold, you will conceive in your womb and bear a son, and you shall call his name Jesus. He will be great and will be called the Son of the Most High. And the Lord God will give to him the throne of his father David, and he will reign over the house of Jacob forever, and of his kingdom there will be no end.”
And Mary said to the angel, “How will this be, since I am a virgin?”
And the angel answered her, “The Holy Spirit will come upon you, and the power of the Most High will overshadow you; therefore the child to be born will be called holy — the Son of God.

-- Luke 1:31-35 (ESV)

Nathanael

Philip findeth Nathanael, and saith unto him, "We have found him, of whom Moses in the law, and the prophets, did write, Jesus of Nazareth, the son of Joseph."
And Nathanael said unto him, "Can there any good thing come out of Nazareth?"
Philip saith unto him, "Come and see."

Jesus saw Nathanael coming to him, and saith of him, "Behold an Israelite indeed, in whom is no guile!" Nathanael saith unto him, "Whence knowest thou me?"
Jesus answered and said unto him, "Before that Philip called thee, when thou wast under the fig tree, I saw thee." Nathanael answered and saith unto him, "Rabbi, thou art the Son of God; thou art the King of Israel. Jesus answered and said unto him, "Because I said unto thee, 'I saw thee under the fig tree', believest thou? thou shalt see greater things than these."
And he saith unto him, "Verily, verily, I say unto you, Hereafter ye shall see heaven open, and the angels of God ascending and descending upon the Son of man.

-- John 1:45-51 (KJV)

The centurion

When the centurion and those who were with him, keeping watch over Jesus, saw the earthquake and what took place, they were filled with awe and said, “Truly this was the Son of God!” -- Matthew 27:54 (NKJV)

The devil

The devil never requested Jesus to proof that "he is God", but instead he challenged Jesus to proof that he is "the Son of God":

The devil said to him, “If you are the Son of God, command ...”
And he (the devil) took him (Jesus) to Jerusalem and set him on the pinnacle of the temple and said to him, “If you are the Son of God, throw yourself down from here...

-- Luke 4:3,9 (ESV)