Some Trinitarians try to claim that the last half of the word monoGENES "only-BEGOTTEN" is not from ginomai ("to come into being" ['born']) but from genos ("kind"). Hence, they claim, the term refers to "the only one of a class or kind." Thus some trinitarian translations speak of Jesus as the "only Son" (see RSV, NEB, JB, AT quotes at beginning) rather than the "only-begotten Son" of God (John 1:14; 3:16, 18; 1 Jn 4:9) - KJV, ASV, NASB.
However, even if we accept the claim that genos is the correct source word for monogenes, we need to examine the claim of some trinitarians that genos does not include the meaning of "begotten"/"made." The Greek word genos has "offspring" and "birth" as some of its meanings even in the trinitarian NT Concordances (Young's Analytical Concordance of The Bible; Strong's Exhaustive Concordance of the Bible; and New American Standard Exhaustive Concordance of the Bible, p. 1640).
The very trinitarian W. E. Vine in his highly-regarded An Expository Dictionary of New Testament Words, p. 805, ADMITS that genos is "(akin to ginomai, to become), [and] denotes an offspring."
Yes, even the trinitarian RSV and NEB (noted above as rendering monogenes as "only" in certain verses relating to Jesus - including Jn 1:18) were forced to use the proper meaning of "offspring" for genos itself at Rev. 22:16 - "I Jesus....am the root and offspring [genos] of David." Compare Acts 17:28, 29 - "'For we indeed are his offspring [genos].' Being then God's offspring [genos], we ought not to think that the Deity is like gold, silver or stone..." - RSV.
According to certain trinitarians, then, the above Scriptures plainly state that Jesus must be one of the kind [genos] of David [or of the David kind]- Rev. 22:16, and Christians and non-Christian Athenians must be of the God kind [genos] - Acts 17:28, 29. This is obviously ridiculous and the proper meaning of "begotten" or "made/produced" cannot be avoided in these Scriptures. Christians (and the men of Athens whom Paul was speaking to) were made or created by God and are His genos ("offspring" or "begotten") in that sense.
Anything that is "begotten" or "born" (or a "son"), then, is something that at one time did not exist and then was brought into existence. (E.g., Adam, the creation of God was called the "SON of God" - Luke 3:38.) This does not refer simply to Jesus' earthly existence but also to his original heavenly existence as shown by 1 John 4:9 which refers to the time when Jesus was "in the beginning with God," even "before the world was." - (John 1:1, 2; 17:5, 24). At that time he was already "the only-begotten [monogenes] Son." - 1 John 4:9, NASB, ASV, KJV. Even the highly trinitarian NT Greek scholar, W. E. Vine, in his An Expository Dictionary of New Testament Words, p. 813, admits that Jesus was the Father's "only-begotten Son" before he came to earth.
For much more, see:
Only-begotten god (OBGOD) (Examining the Trinity)
Only-Begotten (Monogenes) - Links to Information (Search For Bible Truths)