Answer: μονογενοῦς is a compound of two parts, μονο and γένος, literally, "One Offspring/Kindred/Race ... However, given the contexts, and extant literature, it appears that this term has a specific connotation--a sense of Birthright and Inheritance, not necessarily the "only child."
Flavius Josephus, Isaac the Only Begontten??
book 1, section 222: Ἴσακον δὲ ὁ πατὴρ Ἅβραμος ὑπερηγάπα μονογενῆ ὄντα καὶ ἐπὶ γήρως οὐδῷ κατὰ δωρεὰν αὐτῷ τοῦ
Now Abraham greatly loved Isaac, as being his only begotten and given to him at the borders of old age,
But, Abraham didn't have just one child, from his own loins--if we consider Ishmael.
Answer: Therefore, it must be inferred that there is a Connotation at work here, along with the Denotation--specifically the sense/connotation of sole Inheritor, or Birthright.
The Connotation of Birthright Can also be seen in Plato:
Laws, Book 3, [691ε] ἐκ μονογενοῦς, εἰς τὸ μέτριον μᾶλλον συνέστειλε.
Laws, Book 3, [691e] your kingly line no longer single but twofold. In the next place, some man, 2 in whom human nature was blended with power divine, observing your government to be still swollen with fever, blended the self-willed force
*Kingly is inferred from the Context.
Other Children being Born γενος of God, in this Context:
In addition, John speaks of other Children of God, those born of the Spirit, inheriting the nature of God--but he distinguishes the first son, as the heir with this specific Greek term.
First, it should be noted that "Theos," or "God," in Greek, is NOT a name, it is a Nature/Genus/Species, properly translated as "divinity." This "God nature," or rather, "Spirit," (for God is Spirit, John 4:24), is a nature handed down to Children, (that which is born of the Spirit is Spirit).
Psalms 82:6 - I said, “You are gods, And all of you are sons of the Most High.
John 10:34 - Jesus answered them, “Is it not written in your law, ‘I said, “You are gods”’? 35 If He called them gods, to whom the word of God came (and the Scripture cannot be broken), 36 do you say of Him whom the Father sanctified and sent into the world, ‘You are blaspheming,’ because I said, ‘I am the Son of God’?
John goes to great lengths to play on this root, to convey this idea :
John 1:12 - Ὅσοι δὲ ἔλαβον αὐτόν, ἔδωκεν αὐτοῖς ἐξουσίαν τέκνα θεοῦ γενέσθαι, τοῖς πιστεύουσιν εἰς τὸ ὄνομα αὐτοῦ·
NASB: But as many as received Him, to them He gave the right to become children of God, even to those who believe in His name,
John 1:14 - Καὶ ὁ λόγος σὰρξ ἐγένετο, καὶ ἐσκήνωσεν ἐν ἡμῖν― καὶ ἐθεασάμεθα τὴν δόξαν αὐτοῦ, δόξαν ὡς μονογενοῦς παρὰ πατρός― πλήρης χάριτος καὶ ἀληθείας.
NASB: And the Word became flesh, and dwelt among us, and we saw His glory, glory as of the only begotten from the Father, full of grace and truth.
John 3:5 - Ἀπεκρίθη Ἰησοῦς, Ἀμὴν ἀμὴν λέγω σοι, ἐὰν μή τις γεννηθῇ ἐξ ὕδατος καὶ πνεύματος, οὐ δύναται εἰσελθεῖν εἰς τὴν βασιλείαν τοῦ θεοῦ.
NASB: Jesus answered, “Truly, truly, I say to you, unless one is born of water and the Spirit he cannot enter into the kingdom of God.
John 3:6 - Tὸ γεγεννημένον ἐκ τῆς σαρκὸς σάρξ ἐστιν· καὶ τὸ γεγεννημένον ἐκ τοῦ πνεύματος πνεῦμά ἐστιν.
NASB: That which is born of the flesh is flesh, and that which is born of the Spirit is spirit.
John 4:24 - Πνεῦμα ὁ θεός· καὶ τοὺς προσκυνοῦντας αὐτόν, ἐν πνεύματι καὶ ἀληθείᾳ δεῖ προσκυνεῖν.
God is spirit, and those who worship Him must worship in spirit and truth.”