The problem is something no lexicon can offer an answer to, and that is because the issue here is with the syntax/sentence structure, not in terminology.

The passage in question is not simply implying that the Logos is “divine” or “god-like” as often espoused by Jehovah's Witnesses, because John does not use the adjectival (θεῖος [“divine”]), but rather uses the noun form (θεὸς [“God”]); however, that is not to say that nouns cannot, within their semantic domain, convey qualities. Take for example, John 3.6,

"That which is born of the flesh is flesh, and that which is born of the Spirit is spirit."

The idea here has absolutely nothing to do with identification of any sort (“the spirit,” “a spirit”), but everything to do with that of predication. More specifically, the nouns (“flesh,” “spirit”) here function in a purely qualitative sense, without a definite or indefinite semantic force. The context of the passage in view is about the inherent nature of sinful flesh (John 3.6a) in contrast to the new nature of man in the process of regeneration (John 3.6b). Likewise, a similar idea being portrayed is found in 1 John 1.5 (“God is light; in Him there is no darkness at all”), where it is God's essence and nature that is being described in contrast to “darkness.” That is, God has all the qualities, and attributes of light — He is just, holy, and good — therefore, light is an attribute/characteristic inhering within God. Further examples include, but are not limited to, John 6.63 (“The words I have spoken to you are spirit and they are life”), 1 John 4.8 (“…because God is love”), Matthew 12.8 (“For the Son of Man is Lord of the Sabbath” [not “the Lord of the Sabbath,” or “a Lord of the Sabbath”]), et al.

With that being said, the Trinitarian approach to John 1.1 does not emphasize the identity of the Word (thus, the reason for the anarthrous θεὸς), but stresses the nature of the Word. Call attention to what Henry Alford, a 19th c. Anglican theologian wrote in his commentary on this passage,

“The omission of the article before θεὸς is not mere usage; it could not have been here expressed, whatever place the word might hold in the sentence. ὁ λόγος ἦν ὁ θεὸς would destroy the idea of the λόγος altogether. θεὸς must then be taken as implying God, in substance and essence, -not ὁ θεὸς, ‘the Father,’ in Person. . . . as in σὰρξ ἐγένετο [John 1.14], σὰρξ expresses that state into which the Divine Word entered by a definite act, so in θεὸς ἦν [John 1.1c], θεὸς expresses that essence which was His ἐν ἀρχῇ [“In the beginning”]: -that He was very God.”—Henry Alford

The point Alford is driving at here in his comparison of vv. 1 (θεὸς ἦν), and 14 (σὰρξ ἐγένετο), is not only are the two parallel passages conveying similar thought, but John’s placement of the noun before the verb in each passage is significant in that it stresses the qualities or nature of the subject. The positioning of θεὸς before the verb ἦν is what is known as a preverbal predicate nominative. Since John has identified ὁ λόγος (“the Word”) as the subject of the verse, this means that θεὸς in John 1.1b is a subject complement which further identifies the subject. In other words, θεὸς serves to describe the nature and essence of the Word, and this is precisely what we as Trinitarians believe. Not that the Word’s identity is being stressed, rather, it is the intrinsic nature of the Logos that is being portrayed here. All the qualities, attributes, and nature of God — everything that makes God, God — the Word also possesses. This text then, is teaching the equality of nature between the Father, and the Son (c.f. Hebrews 1.3).

Further illustrations can be found in the following statements: (1) “Eve was with Man, and Eve was Man,” and (2) “Helen Reddy sang, ‘I am Woman.’” In both of these statements Eve is not to be equated with the one whom she is “with,” nor is Helen equated with “Woman,” rather, the qualities, characteristics, and nature of “Man” are predicated to Eve (thus, Eve is fully human), and all the qualities of “Woman” are fully predicated to Helen. In this same sense, the qualities of God are fully predicated to the Word in John 1.1c.

What I think should be pointed out is that though Jehovah’s Witnesses often cite Moffatt’s translation of the New Testament (which does translate John 1.1 adjectively, “the Word was divine”) in support of the NWT’s rendition of the text (“the Word was a god”), but what they often fail to understand is that these translations underscore the Trinitarian understanding of the text. In Moffatt’s own words:

“'The Word was God... And the Word became flesh,' simply means the Word was divine... And the Word became human.' The Nicene faith, in the Chalcedon definition, was intended to conserve both of these truths against theories that failed to present Jesus as truly God and truly man..." (Jesus Christ the Same [Abingdon-Cokesbury, 1945], p. 61

Though, this is not comprehensive, one of the methods John could have used to clearly portray the Logos as “a god” would have been through placement of the verb ἦν before the anarthrous predicate (ὁ λόγος ἦν θεὸς [“the Word was a god”]), compare with Acts 28.6 and John 1.6, and the placement of the verbs in each passage,

a. αὐτὸν εἶναι θεόν (Acts 28.6, “he was a god”)

b. Ἐγένετο ἄνθρωπος (John 1.6, “There was a man”)

c. ὁ λόγος ἦν θεὸς (“the Word was a god”)

Alternatively, while Koine Greek does not possess an indefinite article, John could have even used εἷς, which can, and does at times function as an indefinite article, (see Bauer, Arndt and Gingrich: A Greek-English Lexicon of the New Testament and other Early Christian Literature states, “εἷς can mean exactly the same thing as the indefinite article” [pg. 231]) in order to indicate that “the Word was a god,” or “the Word was one of the gods,” see Matthew 8.19, Matthew 26.69 (μία, nominative singular feminine of εἷς), Revelation 8.13 (ἑνὸς, genitive singular masculine of εἷς), et al. That, or John could have made use of the indefinite pronoun τις to indicate that the Word was “a certain” god, but not the one he was referring to in John 1.1b (c.f. Mark 14.51, Luke 8.27, Luke 1.5, and Luke 11.1).

So, while there are other methods (and this is by no means a comprehensive list) John could have used to portray indefiniteness, the same cannot be said, however, should John had wanted to portray the Logos as One who is distinct from God the Father (John 1.1b), while maintaining that the Logos shares, or possesses all the qualities which make God (“the God” of John 1.1b included), God, so that there is no ontological shift between the Father, and the Word.

Since John 1.1, and John 1.14 both have the same grammatical structure (anarthrous predicate noun preceding the verb), and they occur so closely together, John intends us to take them as parallel in meaning, being semantically identical to one another,

a. θεὸς ἦν (John 1.1, anarthrous predicate noun θεὸς preceding the verb ἦν)

b. σὰρξ ἐγένετο (John 1.14, anarthrous noun σὰρξ preceding the verb ἐγένετο)

While John1.14 describes the incarnate nature of Christ, in that He possesses all the fleshly attributes, and qualities of Man; John 1.1 describes Christ’s pre-incarnate nature, insomuch that He possesses all the attributes, and qualities of God. The Word became human, not simply “a man.” The result of Christ being human is that He is a man, but that is not what the phrase is stating. It is not referring to His membership in a class but rather His characteristics that He attained in the incarnation. With that being said, should John had wished to indicate to us that the Word “became a man,” he could have done so as he did eight verses earlier of John the Baptist,

John 1.6, EGENETO ANTHROPOS (“There came a man”)

Also, notice the verb placement as compared to the verb placement of John 1.14, SARX ENGENETO (“became flesh”). John's point is to emphasize the qualities of humanity that were added to the nature of the Word. The semantic connection of SARX (Q-d) to the semantic notion of THEOS (Q) is unmistakable in the discourse.