Posted this in a comment, but it took me 3 hours so I'm posting it separately too just in case anyone else would like to see or comment.
Οὕτως γὰρ ἠγάπησεν ὁ θεὸς τὸν κόσμον, ὥστε τὸν υἱὸν τὸν μονογενῆ ἔδωκεν, ἵνα πᾶς ὁ πιστεύων εἰς αὐτὸν μὴ ἀπληται ἀλλ᾽ ἔχῃ ζωὴν αἰώνιον
Which is the following in the NIV:
For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life.
and NRSV:
For God so loved the world that he gave his only Son, so that everyone who believes in him may not perish but may have eternal life.
I'd argue a more accurate translation would be something like:
.
.
.
... which is to say, in this manner, the god loved the entire created order of the cosmos with a discriminating affection, he produced the one uniquely begotten son and heir for the intended result that all and every, the indwelled trusting into and unto him, have not annihilation, but on the contrary, possess eternal fullness of living.
.
.
.
I think what's especially key with this verse, is that it doesn't say "God sent his son to die because he had emotions for human beings and he wanted them to live forever at some future time," as it is often cited to mean. It says something much larger, more encompassing, and astonishingly beautiful. It is about a present reality, not a future, is is for the entire cosmos, not just humans.
Details posted in comments.
What do you think?
The OP requests clarification about why the English is not:
?For God so loved the world, that he gave the only son....
What is the meaning of this English, and does it accurately convey the Greek? To me, this construction is questionably intelligible. It seems to imply that there was never another son (of anyone), which is patently false, causing me to pause and wonder what is being conveyed and....I don't know exactly.1 This remains an appropriate translation if the Greek carries that aspect of questionable intelligibility. I don't think it does.
The Greek article may function as a possessive pronoun.2 It regularly does so when the antecedent is obvious. Based on both the local and the larger context, I think this is the case in John 3:16.
Locally
This sentence begins a short description of “his son” that maintains the consistent antecedent, continuing through verse 18. There it concludes with a restatement of “the son” using a genitive specifier, “of God”. In case there was any doubt that this is the same son, it also repeats the adjective μονογενής:
οὕτως γὰρ ἠγάπησεν ὁ θεὸς τὸν κόσμον, ὥστε τὸν υἱὸν τὸν μονογενῆ ἔδωκεν...
For God so loved the world that he gave his only son...οὐ γὰρ ἀπέστειλεν ὁ θεὸς τὸν υἱὸν εἰς τὸν κόσμον ἵνα κρίνῃ τὸν κόσμον...
God did not send his son into the world to condemn the world...ὁ δὲ μὴ πιστεύων ἤδη κέκριται, ὅτι μὴ πεπίστευκεν εἰς τὸ ὄνομα τοῦ μονογενοῦς υἱοῦ τοῦ θεοῦ.
He who does not believe is already condemned because he has not believed in the name of the only son of God.
Whose son is now explicit.
More broadly
I don’t think there is much controversy that John’s gospel has at its focus a man he calls the son of God. John the Baptist encapsulates this at the outset (1:34):
κἀγὼ ἑώρακα καὶ μεμαρτύρηκα ὅτι οὗτός ἐστιν ὁ υἱὸς τοῦ θεοῦ.
And I have seen and have testified that this is the son of God.
And the author clearly shares this assessment (20:31):
ταῦτα δὲ γέγραπται ἵνα πιστεύ[σ]ητε ὅτι Ἰησοῦς ἐστιν ὁ χριστὸς ὁ υἱὸς τοῦ θεοῦ...
These things are written so that you might believe that Jesus is the Christ, the son of God...
A twist
This would settle it in my mind but for the fact that Gospel of John often refers to this same individual using the unmodified the Son,3 otherwise odd English. This title (?), though, is used in John’s Gospel only in close association with the Father, an equally odd expression outside of Biblical English. In 3:16, we have instead God. Also, the Son without a possessive modifier generally doesn’t take any modifiers, another reason it would be difficult to read 3:16 as the only Son.
When John’s Gospel speaks of Jesus in relation to God, it is his son.4
1. The reason this seems odd, to my reckoning, is that the semantics of son (and, equally, υἱὸς) requires reference to a father. When definite, English generally uses a possessive construction to contain both required aspects of the concept: "his son" or "Bob's son" or "the son of Bob". "The son ∅" is occasionally used in contexts where the referent and his father are specified in relation to each other but remain abstract and not unique ("If the father dies, the son will inherit...”). This is clearly not the case here.
2. For an example in what might be an "easier" language, if you know Spanish (and probably similarly in other post-Latin languages with a definite article, not sure), you would say, Abro los ojos and nobody would wonder whose eyes. The possessive pronoun would generally be used only if the subject and the possessor of the eyes were two different individuals. (Native speakers should feel free to correct me.) In English, this usage of the article would be strange.
3. I throw my hands up about the capitalization decisions of various translations here.
4. This is a bit (OK, completely) circular, but it's consistent with English idiom (as realized in the Gospel of John).
The phrase appears to be correctly rendered in all of the above translations that you have listed. For example, Young's Literal Translation renders John 3:16 thus:"for God did so love the world, that His Son -- the only begotten -- He gave, that every one who is believing in him may not perish, but may have life age-during" That God the Father 'gave' [didōmi] 'His' [autos] 'only begotten' [monogenēs] Son [huios] is in the text.