According to Jehovah’s Witnesses why did John write the words found in John 1:1? Jehovah's Witnesses translate John 1:1 as: 

" In the beginning was the Word, and the Word was with God, and the Word was a god."

whereas its more generally translatedas something like :

"In the beginning was the Word, and the Word was with God, and the Word was God."

i.e. the NWT adds the word "a". This is justified because the Greek language of the first century did not have an indefinite article, and they feel that this rendering of the verse makes it more consistent with the rest of the bible (which they see as teaching that Christ is an Archangel, rather than a person within the Godhead).

If this is the case, why do they think John used this phrase? That is what was he trying to communicate? 

If John is simply trying to say that Christ was with God, and through him all things were created, this clause could be removed from the sentence- and the passage would become clearer, because Johns Greek speaking audience wouldn't have to wonder if John means Jehovah or simply some lesser god.   

Additionally this would avoid all the problems that they see as coming from the miss-translation of the verse being used to support trinitarianism

EDIT: 

This Question is flagged as similar to another. The difference is subtle- that one asks what textual support is there for the verse, this asks, assuming that the JW understanding of the scriptures and Johns intentions are true, why John would write the verse as it is - appearing to endorse at least the divinity of Christ.
The reason why is because they like other Christians believe that the context of John's whole gospel is Jesus and his role in salvation. One JW said it this way to me: John wanted to establish Jesus as his agent for creation as well as his instrument for salvation. He wanted to make it clear that Jesus existed in the presence of God as an angel or (lesser god) before he came to be a man on earth.

Now, you're right. It certainly is peculiar for a first century Jew to use words akin to hierarchal duo-theism while his contemporaries were strictly monotheistic. Why didn't he just say out right that he was an angel? That certainly would have cleared up the issue. John like other jews believed in one God and that no other god was formed after him or before him (Isa. 43:10) making this interpretation of John 1:1 problematic. 

Having debated many JW's myself over this verse, they find comfort in their logic for the lack of the article making theos indefinite rather than definite, but when applying the same logic to other indefinite thoes's in the same chapter and even in the same book, they do not apply the same logic. In other words their are many places where thoes lacks the definite article but they do not render those passages as 'a god'. Why the inconsistency? In my opinion, it's theological bias.