From Anstey’s Doctrinal Definitions:
This is a term of endearment that an only child has in the affections of his father (Luke 8:42; John 1:14; 3:16, 18; 1 John 4:9) or mother (Luke 7:12). Begetting, in the sense in which the word is used in this term, does not refer to a person's congenial beginning—his birth. A clear proof of this is that the Lord was “the only-begotten Son” before He was born into this world (John 3:16). The emphasis in the term is on "only," rather than on “begotten.” Christ is the Father's one and only-begotten Son. Loosely translated, it could read: "dearly beloved One."
Hence, when the term “Only-begotten” is applied to the Lord Jesus, it is referring to His uncreated relationship with God the Father as His dearly beloved Son. It denotes the Father’s delight in Him. John 1:14 speaks of the glory that men beheld in the Lord when they saw Him living in the enjoyment of His Father’s love. John said, in a parenthesis, that it is similar to that which an only-begotten child has with his father, having his father’s full and undivided attention and affection. (This is why "only-begotten with a father" is not capitalized in the text; it is referring to the human relationship of a father with his son, and the Spirit of God is using it to illustrate the affection the Father has for the Son.) Thus, the Lord was the Object of His Father’s undivided attention and delight (Matt. 3:17), for He always dwelt “in the bosom of the Father” as “the only begotten Son” (John 1:18; Prov. 8:30) and “the Son of His love” (Col. 1:13).