The expression “only begotten” with reference to Jesus Christ as the Son of God occurs five times in the KJV New Testament (all in the Johannine literature: John 1:14 and 18, 3:16 and 18, and 1 John 4:9), nine times in the Book of Mormon (beginning with 2 Nephi 25:12), 13 times in the Doctrine & Covenants (beginning with D&C 20:21), and a remarkable 25 times in the Book of Moses. (Joseph also added the expression to 1 Timothy 2:4 in the JST: “who is the Only Begotten Son of God.”) The New Testament preserves four other uses of the word where it is not predicated of Jesus: a widow’s son (Luke 7:12), Jairus’ daughter (Luke 8:42), the son of an unnamed man in a crowd (Luke 9:38), and Isaac, as the son of Abraham (Hebrews 11:17). It also appears a number of times in the Septuagint, and in classical literature beginning with Hesiod.
The expression “only begotten” in the KJV is a mistranslation of monogenes, which derives from monos “only” and the root GEN, which can refer to either descent or type. Note, for instance, that the English words “kin” and “kind” both derive from this same Greek root. Thus, if in context monogenes means the only number of a kin, the sense is “sole descent, the only child of one’s parents.” Conversely, if in context the word means the only one of a kind, in a qualitative sense, the connotation is “only, peerless, matchless, unique, of singular importance, the only one of its kind.” If the word specifically were meant to convey the connotation “only begotten,” it would be monogennetos, from the verb gennao “to beget.” The fact that monogenes has only one “n” in the second half of the word and not two is a clear indication that it is not derived from gennao.
The Old Latin manuscripts had translated monogenes correctly into Latin with unicus, meaning “only,” but Jerome in the Vulgate generally changed this translation to unigenitus, “only begotten.” This appears to have been influenced by the Orations of Gregory of Nazianzus, who, in discussing the eternal relation between the Father and the Son, spoke of the Father as gennetor “begetter” and the Son as gennema “begotten.” This was a response to the Arian argument that Jesus was not begotten, but rather made.
William Tyndale, the first to translate the New Testament from Greek into English, corrected this in some (but not all) passages with “only sonne.” The KJV, however, showing the strong influence of the Vulgate, reverted to “only begotten son,” which became the normative English rendering until the rise of modern translations.
John did not refer to Jesus as God’s “only begotten” son, because he designated all who received Jesus as being spiritually begotten of God. See, for example, John 1:12-13:
“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: which were born [i.e., begotten], not of blood, nor of the will of the flesh, nor of the will of man, but of God.”
I have always found the expression of this concept in 1 John 3:9 quite striking:
“Whosoever is born [begotten] of God doth not commit sin; for his [i.e., God's] seed [Greek sperma] remaineth in him: and he cannot sin, because he is born [begotten] of God.”
The use of “only begotten” in the Book of Mormon, Doctrine & Covenants and Book of Moses is simply a reflection of the English translational tradition. We are not particularly concerned with the theological battle that gave rise to this translation many centuries ago. Although it is a mistranslation, it is not that far removed from the correct meaning, as the suggestion of being begotten is to some extent inherent in the designation “son” (huios) itself.