Monogenes

What is meant by Jesus being God the Father's “only” (John 1:18, 3:16 and 3:18) Son?
The original Greek adjective μονογενης (monogenes) in John 1:18 and its grammatical case variants (slightly different endings to describe the object of a verb vs. to describe the subject of a verb, etc.) μονογενη (monogene) in John 3:16, and μονογενους (monogenous) in John 3:18, are all derived from μονος (monos), which means “alone” or “solitary,” and γενoς (genos), which means “type” or “kind”; this adjective literally means “alone type” or “solitary kind” - i.e., “unique” or “only” - and when used to describe the “Son” (John 1:18, 3:16, 3:18), describes Him as the “unique” or “only” Son of God the Father. (The case variant μονογενους (monogenous) is also in John 1:14 but as a substantive, which is an adjective that replaces a noun instead of describing it (e.g., in the phrase, “the land of the free, and the home of the brave,” “free” and “brave” are substantives), and therefore means “only Son” on its own, without the noun for “Son.”)

Can monogenes, monogene and monogenous be translated as “only begotten” to describe Jesus as the “only begotten Son” of God the Father?
Jesus is “God” (John 1:1), and the Bible says God is “eternal” (Deuteronomy 33:27). “Begotten” means “having brought forth” or “having caused to arise.” Anyone “begotten” in the sense of having been caused to exist did not exist until he was begotten, so he wouldn't be “eternal,” which is defined as having no beginning and no end (see Everlasting life). Therefore, “only begotten” would be a mistranslation that contradicts what the Bible says about Jesus' deity.

But doesn't Acts 13:33 quote God the Father as saying that He has “begotten” Jesus?
γεγεννηκα (gegenneka), the original Greek word translated “have begotten” in Acts 13:33, is different from μονογενης (monogenes), and God the Father is talking about bringing forth Jesus from His grave after His death on the cross (see “Begotten” meaning).

Then why do some English translations translate monogenes, monogene and monogenous as “only begotten”?
Latin, the official language of the Roman Empire, was the first language into which the New Testament, originally written in Greek, was translated, and the initial translation into Latin correctly translated all nine occurrences of μονογενης (monogenes) and its case variants in the New Testament as unicus, which means “only” or “unique” (the English word “unique” comes from this Latin word). But when a 4th century heretic named Arius began to spread a heresy that Jesus must have been created (see My Father is Greater than I), Jerome, a prominent figure back then in the budding Roman Catholic church, somehow decided “begotten” is needed to fight off “created,” and changed unicus to unigenitus, which means “only begotten,” for six of the nine occurrences, including all five that refer to Jesus (John 1:14, 1:18, 3:16, 3:18, 1 John 4:9; the sixth was Hebrews 11:17 about Isaac), while leaving unicus alone in Luke 7:12, 8:42 and 9:38. Jerome's translation of the Bible into Latin, called Latin Vulgate, remained the official Bible of Roman Catholicism until 1979, and some English translations still pay homage to it.

Is that homage warranted?
English versions of the New Testament need to translate the original Greek directly into modern English, and the way to stamp out a heresy is to simply present what the Bible says, not to change it or to create another heresy; the antidote to falsehood isn't a lesser falsehood, but the Truth.

Then why is Jesus called the “Son” of God the Father?
See Son of God.