Summary
μονογενής is only begotten but not in the sense of normal procreation. This is obvious from the text: the μονογενής is begotten γίνομαι, not begotten γεννάω1
. It is also obvious from the context. John, who alone calls Jesus μονογενής, states one must be γεννάω again to become, γίνομαι a child of God. Thus a critical difference between the μονογενής who is the Son of God and another μονογενής, is that the Son is from the beginning of God where all others begotten γεννάω or only begotten μονογενής, must be begotten γεννάω again to be, γίνομαι of God.
The essence of μονογενής in John is on future ancestors: the Son of God, the one who is and by whom all will become children of God; not a μονογενής like Isaac, unique because past ancestry.
The Greek Text
Ignoring those translations which have purposely chosen an approach such as an amplified or paraphrased reading, here are reasons translations of the New Testament have different readings:
- Assumption of which text best represents the original
- Translate a word differently into English
- Translation philosophy (i.e. word-for word or thought-for thought)
As there are no original copies, translators must first choose a text from which to work. The three main choices are Textus Receptus (TR) used by the King James family, the Majority/Byzantine Text (MT) used primarily by the Orthodox Church, and Novum Testamentum Graece (GNT) used in most modern translations. Generally there are only minor variations and that is case at John 3:16:
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. (AKJV)
Οὕτως γὰρ ἠγάπησεν ὁ θεὸς τὸν κόσμον ὥστε τὸν υἱὸν αὐτοῦ τὸν μονογενῆ ἔδωκεν ἵνα πᾶς ὁ πιστεύων εἰς αὐτὸν μὴ ἀπόληται ἀλλ᾽ ἔχῃ ζωὴν αἰώνιον (TR)
“For God so loved the world that He gave His only-born Son, in order that everyone believing in Him may not perish, but may have eternal life. (DLNT)
οὕτως γὰρ ἠγάπησεν ὁ θεὸς τὸν κόσμον ὥστε τὸν υἱὸν [-------] τὸν μονογενῆ ἔδωκεν ἵνα πᾶς ὁ πιστεύων εἰς αὐτὸν μὴ ἀπόληται ἀλλ᾽ ἔχῃ ζωὴν αἰώνιον (GNT)
“For God so loved the world that He gave His only begotten Son, that whoever believes in Him should not perish, but have eternal life. (MEV)
ουτως γαρ ηγαπησεν ο θεος τον κοσμον ωστε τον υιον αυτου τον μονογενη εδωκεν ινα πας ο πιστευων εις αυτον μη αποληται αλλ εχη ζωην αιωνιον (MT)
The only difference is the omission of αὐτοῦ (his) in the GNT, which as the DLNT shows, is implied and so is added. Essentially there is no conflict in the Greek texts. Additionally, it is significant that Jehovah Witnesses, who reject the divinity of Jesus, have the same understanding of the verse:
"For God loved the world so much that he gave his only-begotten Son, in order that everyone exercising faith in him might not be destroyed but have everlasting life. (NWT)
Therefore, if there is a question on the divinity of Jesus arising from John 3:16, it stems from how μονογενής is translated.
Only "Begotten"
Only "begotten" is how the AKJV and others translate μονογενής, which is from μόνος ("only") and γίνομαι ("begotten"). It is used in context with Jesus 5 times, all by John: 4 in the Gospel and once in the first Letter. The issue is deciding how γίνομαι should be translated. According to Thayer's, it is a verb meaning to become, most often translated as "be" (KJV). Before first using μονογενής, John shows the distinction between the two verbs "begotten" γίνομαι and "begotten" γεννάω:
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: which were born, not of blood, nor of the will of the flesh, nor of the will of man, but of God. (1:12-13) [AKJV]
ὅσοι δὲ ἔλαβον αὐτόν ἔδωκεν αὐτοῖς ἐξουσίαν τέκνα θεοῦ γενέσθαι τοῖς πιστεύουσιν εἰς τὸ ὄνομα αὐτοῦ οἳ οὐκ ἐξ αἱμάτων οὐδὲ ἐκ θελήματος σαρκὸς οὐδὲ ἐκ θελήματος ἀνδρὸς ἀλλ᾽ ἐκ θεοῦ ἐγεννήθησαν
The only-γίνομαι, is not only-γεννάω. Rather, humans who were already born (γεννάω) either of blood(s), or of the will of the flesh, or of the will of man may become (γίνομαι) children of God. John 3:3-8 describes this as being born again, or born from above. (See this related question Why is it that in John 3:3 the ESV translates "be born from above" to "is born again"?).
In the NT μονογενής is also used to describe Isaac, a widow's dead son, a dying 12-year old girl, and a demon possessed boy. Yet, regardless if a person is simply begotten or called an only begotten, all have the same requirement to become children of God. On the other hand, the Word who became flesh is the μονογενής of the Father:

John describes the family of God as made up of people who were "begottens" or "only begottens" but have been begotten again and one who is different from all others. The unique one is called μονογενής. In other words, in the family of God only the Son of God is the μονογενής. It may not be correct linguistically, but as it pertains to Jesus, John's use means the only begotten is the "only begetter" of children of God.
To eliminate confusion, I believe the best approach is to leave the word untranslated when used to describe Jesus. This would preserve the unique nature, transliterating as Monogenēs in the Gospel and Letter. In fact, this may be what Jesus is saying in His final use of μονογενής:
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 3:18 AKJV)
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 Monogenēs, Son of God.
"Name" (ὄνομα) can be "used for everything which the name covers, everything the thought or feeling of which is roused in the mind by mentioning, hearing, remembering, the name, i. e. for one's rank, authority, interests, pleasure, command, excellences, deeds, etc."2
For Jesus, Monogenēs is His position and is synonymous with "Son of God."
Unlike contexts where μονογενής is used to describe a human being and should be translated, Monogenēs, defines Jesus. Therefore, μονογενής should be treated just as is Χριστός and Μεσσίας:
Transliterations in John:
מָשִׁיחַ ---------> Χριστός
Χριστός ------> Christ
μονογενής ----> Monogenēs
The Monogenēs
The meaning of γίνομαι must also consider John's initial uses of γίνομαι:
All things were made (ἐγένετο) through him, and without him was not any thing made (ἐγένετο) that was made (γέγονεν). (John 1:3)
"Only begotten" is misleading because the Monogenēs, who became flesh, made all things, including those who become children of God. John's use of the verb "to be" and the aspect of "being" evokes God's revelation of Himself to Moses:
Then Moses said to God, “If I come to the people of Israel and say to them, ‘The God of your fathers has sent me to you,’ and they ask me, ‘What is his name?’ what shall I say to them?” God said to Moses, “I AM WHO I AM.” [a] And he said, “Say this to the people of Israel: ‘I AM has sent me to you.’” (Exodus 3:13-14)
a. Or I AM WHAT I AM, or I WILL BE WHAT I WILL BE
The reason either I AM WHAT I AM, or I WILL BE WHAT I WILL BE are equally correct is Hebrew does not have a word for the present tense of the verb "to be." In other words, there is no Hebrew word for "am" or "is" or "are."3
Greek, however, does not have this limitation:
καὶ εἶπεν ὁ θεὸς πρὸς Μωυσῆν ἐγώ εἰμι ὁ ὤν καὶ εἶπεν οὕτως ἐρεῖς τοῖς υἱοῖς Ισραηλ ὁ ὢν ἀπέσταλκέν με πρὸς ὑμᾶς (Exodus 3:14)
And God said to Moyses, "I am the One Who is." And he said "Thus shall you say to the sons of Israel, 'The One Who is has sent me to you.'" (NETS)
The Hebrew AM or WILL BE was rendered as IS. In his essay, Robert G. Hall notes how John 1:18 may be read in light of how the LXX renders God's self-identification to Moses:4
θεὸν οὐδεὶς ἑώρακεν πώποτε μονογενὴς θεὸς ὁ ὢν εἰς τὸν κόλπον τοῦ πατρὸς ἐκεῖνος ἐξηγήσατο
(John 1:18)
The phrase, μονογενὴς θεὸς ὁ ὢν which taken literally as "...[the] only begotten God the [One] being..." may also be read with the divine name: "...[the] only begotten God, the One Who is..." Transliterating μονογενής and using Gates' alternate reading yields this translation:5
"No one has ever seen God; [the] Monogenēs God, THE ONE WHO IS, has himself led out into the bosom of the Father."
θεὸν οὐδεὶς ἑώρακεν πώποτε μονογενὴς θεὸς ὁ ὢν εἰς τὸν κόλπον τοῦ πατρὸς ἐκεῖνος ἐξηγήσατο (GNT)
The TR text which the AKJV uses has a variant reading of υἱός (Son) in place of θεὸς (God). However, when μονογενής is transliterated as Monogenēs, this variant has no impact on the meaning:
"No one has ever seen God; [the] Monogenēs Son, THE ONE WHO IS, has himself led out into the bosom of the Father."
θεὸν οὐδεὶς ἑώρακεν πώποτε ὁ μονογενὴς υἱός ὁ ὢν εἰς τὸν κόλπον τοῦ πατρὸς ἐκεῖνος ἐξηγήσατο (TR)
Both state the Monogenēs is ὁ ὢν, THE ONE WHO IS and affirm His divinity.
Conclusion
English translations of μονογενής which use "begotten" evoke an image of human birth and procreation. This image, reinforced by phrases such as "Son of God" can lead to a distortion of the meaning of the original language. However, these images are not consistent with the Greek, regardless of which text is used as the best representation of the original. Unfortunately, any decision to translate μονογενής will fall short of capturing the essence as it pertains to Jesus. In this regard, the AKJV is neither better or worse than any other translation and one's choice of which English translation is best should be based on other considerations.
To avoid creating a misleading image, μονογενής can be treated like Χριστός and transliterated, Monogenēs. This will help avoid the potential of misunderstanding the Greek and preserves the uniqueness as it pertains to Jesus. This treatment will not found be found in any translation. Therefore, the reader must be aware of those places where μονογενής is used as a title or position and mark up or otherwise note the best treatment is the transliteration, Monogenēs.
Notes:
1. The English "begotten" throughout the OT to describe human ancestry is γεννάω in the LXX. It is not γίνομαι, the component of μονογενής.
2. Thayer's Greek Lexicon
3. Dennis Prager, Exodus, God, Slavery, and Freedom, Regnery Faith, 2018, p. 44
4. Robert G. Hall, "The Reader as Apocalyptist in the Gospel of John," John's Gospel and Intimations of Apocalyptic, edited by Catrin H. Williams and Christopher Rowland, Bloomsbury, 2013, p. 268
5. Ibid.