What does the phrase 'Son of God' mean?

A number of Christian apologists argue that Messiah Yeshua Ben Yosef (or 'Christ Jesus'), was the Son of God' and that as God is divine, so must this 'son of God' be divine.

The argument that the ‘son of God' implies some degree of divinity rather than ‘chosen-ness’ is a Hellenistic one though, and certainly not how the Hebraic New Testament writers would have understood the term. 

Yeshua actually declared himself the ‘son of man’ rather than ‘son of god’  (while the term ‘son of man’ in the Bible has three different understandings depending on context), the use of ‘son of God’ is not introduced in the Synoptics (except for in Mark 15 where scholars believe it is most likely a redaction).

The phrase ‘son of God’ was a Jewish one reflected in both the Tanakh (for example  "For to what angel did God even say: Thou are my son, today I have become thy father'? - Ps 2:7, and 'I will be a father to him and he shall be a son to me' -2 Sam 7:14) and in inter-testamental Jewish writings such as the DSS fragment 4Q246. In this ‘Son of God’ fragment it says, “he shall be called the Son of the God; they will call him the Son of the Most High…He will judge the earth in righteousness…and every nation will bow down to him…with (God’s) help he will make war, and…[God] will give all the peoples into his power.” (This fragment is actually referring to an evil ruler being called a ‘son of God’, but it shows emphatically that this concept of the children of Israel being ‘sons of God’ was common).

Another Jewish ‘messianic’ type figure, Honi, the circle maker (from the 1^st Century BCE), also a native of the Galilee, made the rains come. 

He did this by making a circle in the sand, entering into it and saying to God: “Lord of the world, your children have turned to me because I am a son of the house before you.  I swear by your great name that I will not move from here until you are merciful to your children."   

At first mere rain drops appeared.  Honi said "Abba I did not ask for this, but for rains sufficient to fill cisterns, ditches and caves." So the rains fell in sheets.  Honi said "Abba I did not ask for this, but for rains of benevolence, blessing and graciousness." The rain came in the form he requested. 

In the Talmud, Honi often addresses God as Abba - father.  So Honi, was known as the `son of the house - the house of God’, that is he was known as a ‘son of God’ and not thought of as divine. – see Babylonian Talmud - Ta’anit 23a. 

In fact, the Tanakh refers to angels (Gen. 6:2, Deut. 32:8), the Israelites (Ex. 4:24; Deut. 14:1.), Kings of Israel (II Sam. 7:4, Ps. 2:7), and Jewish holy men as sons of God. 

In the Book of Proverbs the person being addressed is called Beni - My son. Who is the speaker - "If you will hear My words and obey My commandments." (Prov. 2:1) My commandments can only refer to God. Thus God is referring to the Jewish people as `my son(s)'.

In the pseudapigraphic book the Wisdom of Solomon, (early 1st Century BCE) we read:

"For if the upright man is God's son God will help him and rescue him from the clutches of his enemy.. Let us test him with cruelty and with torture, and thus explore this gentleness of his and put his patience to the test. Let us condemn him to a shameless death since God will rescue him - or so he claims."

Thus it is quite clear that being a ‘son of God’ or even THE ‘son of God’ is no claim to being God or being Divine (as some pre-existent God-man-angel hybrid-type creature).