I believe that our English vocabulary falls short to adequately translate the underlying Greek word used for "only begotten son". The Greek word is something like: mono genes or mono genus. "mono" stands for one. For example where we use the word "monolith", we mean one continuous mass of rock. And genes (pronounced: "gen-ace" means offspring or gendered one, a son or daughter, one who comes forth from our own "seed". The LORD Himself says of Christ in the Psalms (Psalm 2:7) 'Thou art my son, this day have I begotten thee...' Context tells us that "begotten" simply means that the individual has issued forth from a father and as a result of the father's will to "beget" a child. We know of course, that Jesus has eternally existed with God in "concert". He is not a created being. There is an extensive theological discussion as to WHEN He was declared to be God's Son and I will leave this to the more learned.
Anyway, my own translation for mono genes would be something like: unique, one-of-a-kind issue (son). Thus "only begotten son" or "one and only son" could be rendered: "one unique son".
John 1:12 tells us: '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:' In this passage, we learn that we can become sons (Greek: "tek-non") of God by faith. Our son ship, however is not the same as the unique relationship that Jesus shares with the Father. A weak analogy is that Jesus is sort of our "proxy" with the Father. We are seen by the Father through the merits and attributes of Jesus instead of our own righteousness.
I would like to hear an explanation from someone as to the difference of God saying of Jesus, 'I have begotten you...' compared to His declaration that we can become sons of God by faith.