“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.”
Now as to why the Lord Jesus Christ IS the "only begotten son" and the term “Begotten” should not be removed as in some current bibles is explained in the following:.
Without this term in John 3:16, we lose the prophetic link that was intended from the Old Testament.
According to scripture, Jesus (as manifested as an earthly messiah) occurred on a “day”. This is not an "eternal" day.
Psalm 2:7 “I will declare the decree: the Lord hath said unto me, Thou art my Son; this day have I begotten thee.”
The above verse clearly declares that the Son of God is born on a day. Since we know our Lord has no beginning, it must refer to a future fulfillment in a particular sense. When we study the New Testament, we see this has a four and possible five-fold fulfillment.
The birth of Jesus:
1) Luke 1:35 “And the angel answered and said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God.”
John 1:14 “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” .and
The baptism of Jesus:
2) Mark 1:11 “ And there came a voice from heaven, saying, Thou art my beloved Son, in whom I am well pleased.”.
The mount of transfiguration:
3) 2 Peter 1:16 For we have not followed cunningly devised fables, when we made known unto you the power and coming of our Lord Jesus Christ, but were eyewitnesses of his majesty. 17 For he received from God the Father honour and glory, when there came such a voice to him from the excellent glory, This is my beloved Son, in whom I am well pleased. 18 And this voice which came from heaven we heard, when we were with him in the holy mount.
The return of the Lord Jesus and perhaps also looking to the beginning of the eternal state after the 1000 year reign of Christ.
4) All of Psalm 2.
Here we see that of all the fulfillments of God’s decree of His Son, that all are future and begin with the incarnation, that mysterious fleshly birth when our Lord became “Begotten”.
The INTENT of the word BEGOTTEN therefore in John 3:16 is to turn your attention to Psalm 2:7 and see the magnitude of the fulfillment. The whole point of "only begotten" is to affirm Jesus as the Son of God. Fulfilled at his birth, his baptism and when he installed as king at Jerusalem. Using a term as "the one and only" or something similar as in the NIV or ESV means nothing.
1 John 4:3 “And every spirit that confesseth not that Jesus Christ is come in the flesh is not of God:…”
Before Christ’s incarnation he was the “Word” as stated in John 1:1. He was with God and was God which is a mystery to us mere mortals.