The KJV of John 1:18 reads:
No man hath seen God at any time; the only begotten Son, which is in the bosom of the Father, he had declared him.
The term "only begotten Son" is ο μονογηνης υιος, "the unique Son." Many other translations (e.g. NRSV) and scholars (e.g. Bart D. Ehrman, The Orthodox Corruption of Scripture) accept this as being the original reading.
There is a textual variant, however, to this verse, and many early texts (e.g. P66; P75) contain this reading--μονογενὴς θεὸς.
The NET Bible note for John 1:18 (which accepts μονογενὴς θεὸς as original) has a good summary of the evidence:
The textual problem μονογενὴς θεός (monogenes theos, "the only God") versus ὁ μονογενὴς υἱός (ho monogenes huios, "the only son") is a notoriously difficult one. Only one letter would have differentiated the readings in the MSS, since both words would have been contracted as nomina sacra: thus θσ or υσ. Externally, there are several variants, but they can be grouped essentially by whether they read θεός or υἱός. The majority of MSS, especially the later ones (A C3 Θ Ψ ƒ1, 13 Û lat), read ὁ μονογενὴς υἱός. î75 א1 33 pc have ὁ μονογενὴς θεός, while the anarthrous μονογενὴς θεός is found in î66 א* B C* L pc. The articular θεός is almost certainly a scribal emendation to the anarthrous θεός, for θεός without the article is a much harder reading. The external evidence thus strongly supports μονογενὴς θεός. Internally, although υἱός fits the immediate context more readily, θεός is much more difficult. As well, θεός also explains the origin of the other reading (υἱός), because it is difficult to see why a scribe who found υἱός in the text he was copying would alter it to θεός. Scribes would naturally change the wording to υἱός however, since μονογενὴς υἱός is a uniquely Johannine christological title (cf. Joh 3:16, Joh 3:18; 1Jo 4:9). But θεός as the older and more difficult reading is preferred. As for translation, it makes the most sense to see the word θεός as in apposition to μονογενής, and the participle ὁ ὤν (ho on) as in apposition to θεός, giving in effect three descriptions of Jesus rather than only two. (B. D. Ehrman, The Orthodox Corruption of Scripture, 81, suggests that it is nearly impossible and completely unattested in the NT for an adjective followed immediately by a noun that agrees in gender, number, and case, to be a substantival adjective: "when is an adjective ever used substantivally when it immediately precedes a noun of the same inflection?" This, however, is an overstatement. First, as Ehrman admits, μονογενής in Joh 1:14 is substantival. And since it is an established usage for the adjective in this context, one might well expect that the author would continue to use the adjective substantivally four verses later. Indeed, μονογενής is already moving toward a crystallized substantival adjective in the NT [cf. Luk 9:38; Heb 11:17]; in patristic Greek, the process continued [cf. PGL 881 s.v. 7]. Second, there are several instances in the NT in which a substantival adjective is followed by a noun with which it has complete concord: cf., e.g., Rom 1:30; Gal 3:9; 1Ti 1:9; 2Pe 2:5.) The modern translations which best express this are the NEB (margin) and TEV. Several things should be noted: μονογενής alone, without υἱός, can mean "only son," "unique son," "unique one," etc. (see Joh 1:14). Furthermore, θεός is anarthrous. As such it carries qualitative force much like it does in Joh 1:1, where θεὸς ἦν ὁ λόγος (theos en ho logos) means "the Word was fully God" or "the Word was fully of the essence of deity." Finally, ὁ ὤν occurs in Rev 1:4, Rev 1:8; Rev 4:8, Rev 11:17; and Rev 16:5, but even more significantly in the LXX of Exo 3:14. Putting all of this together leads to the translation given in the text.
One conservative Evangelical commentator said the following about μονογενης and John 1:18:
The Lexham Bible renders this phrase as "the one and only, God"; such begs many important theological questions, including if Jesus is the one and only God, then what about the Father and Spirit? That would mean that the person of "Jesus" is exhaustive of "God" to the exclusion of other divine persons. In reality, for John, "the only true God" is exhausted by the person of Father (John 17:3; cf. 20:17; this article on Jesus' divinity in light of John 17:3).
This leads us to conclude that μονογενης denotes "the only member of a kin or kind" . . . But the connotations that μονογενης derives from Johannine usage greatly enrich the epithet or title. Jesus is μονογενης because . . . (2) He is "unique" (1) in relation to the Father, because (i) both before and after his incarnation he was in the most intimate fellowshp with his Father (1:18), (ii) he was the sole and matchless Revealer of the Father's love (John 3;16; 1 John 4:9), and (iii) his origin is traceable to God the Father (John 1:15; cf. 1 John 5:18); and (b) in relation to human beings, because he is the object of human faith, the means of eternal salvation, and the touchstone of divine judgment (John 3:16, 18) . . . following verse 17, verse 18 suggests that John has in mind a contrast between Moses, who was given a vision of God's back (Exod. 33:18-23) or form (TMNH, Num. 12:6-8) but denied a vision of God's face (Exod. 33:20; but cf. 24:9-10), and Jesus Christ, who, sharing the divine nature (θεος) as the only Son (μονογενης), had not simply seen God on one isolated occasion but had always been known intimately as Father (ο ων εις τον κολπον του πατρος). (Murray Harris, Jesus as God: The New Testament use of Theos in Reference to Jesus [Eugene, Oren.: Wipf & Stock, 2008], 87, 93)
The Lexham Bible renders this phrase as "the one and only, God"; such begs many important theological questions, including if Jesus is the one and only God, then what about the Father and Spirit? That would mean that the person of "Jesus" is exhaustive of "God" to the exclusion of other divine persons. In reality, for John, "the only true God" is exhausted by the person of Father (John 17:3; cf. 20:17; this article on Jesus' divinity in light of John 17:3).
In reality, ο μονογενης θεος means "the unique God." If the person of Jesus is exhaustive of "the unique God," then what about the other divine persons of the Godhead? Furthermore, what does this mean for "monotheism"? In this verse, the Father is exhaustive of “God,” while the Son is exhaustive of “unique God.” Simple logic, mathematics, and linguistics requires that, in a real sense, there is more than one God/θεος in John’s theology.
As a result of this realisation, many scholars have been squeemish about accepting this variant due to it being in conflict with the a priori assumption that John was a Unitarian monotheism, per J.A.T. Robinson in his The Priority of John (or even Trinitarian "monotheism"). Note the following from Kittel, ed. Theological Dictionary of the New Testament:
The phrase can only mean:
“[A]n only-begotten God; to render “an only-begotten one who is God,” is an exegetical invention. It can hardly be credited of Jn., who is distinguished by monumental simplicity of expression. An only-begotten God corresponds to the weakening of monotheism in Gnosticism. (TDNT 7:740)
The “problem” dissipates if one accepts the Latter-day Saint doctrine of the Godhead, the concept of “Kingship monotheism,” as defined thusly:
There are many gods, but all of the gods are subordinate to a Most High God to whom the gods give ultimate honour and glory and without whose authority and approval they do not act in relation to the world. (Blake Ostler, Exploring Mormon Thought, vol. 3: Of God and Gods [Salt Lake City: Greg Kofford Books, 2008], 43).
Latter-day Saint Christology would have no issue with the theological implications of John 1:18, nor would it have an issue with the discussion of this phrase as found in BDAG:
μονογενὴς θεός (considered by many the orig.) an only-begotten one, God (acc. to his real being; i.e. uniquely divine as God’s son and transcending all others alleged to be gods) or a uniquely begotten deity (for the perspective s. J 10:33-36)
In spite of claims to the contrary (see this post on the grammar of John 1:1c), the Gospel of John is a solid witness to the truthfulness of Latter-day Saint teaching on Christology, as well as other topics, such as water baptism being salvific.