What prompted the second century writers to adopt the concept of οὐσία to refer to God's nature? My question When did the concept of οὐσία begin to refer to God's nature? has been answered and the answer was actually great. However, the answer has led me to ask another question. 


Conceptually, I do hold onto the applicability of οὐσία to refer to
God's nature since nature is what innately belongs to someone.
I agree that the Greek word homoousios (same substance) faithfully
preserve the meaning of Scripture in creedal form (Nicene Creed cf.
John 1:1, 18 10:28-33; Hebrews 1:3).


What prompted the second century writers to adopt the concept of οὐσία to refer to God's nature?


Did the second century church writers adopt οὐσία to refer to God's nature because they wanted to summarize what the Bible has to say on the relationship of the Father and the Son? 
Or did they adopt it because they were trying either to defend the Christian faith to Greeks or to convert contemporary Greeks ?
This seems to have been prompted by Greek Philosophy beginning about 600 BC. This is most pronounced in the teachings of Plato, specifically his Theory of Forms in Plato's cave allegory.

Details

In Plato's theory, εἶδος (eidos) represents the "Visible Form" while, μορφή (morphē) represents the "shape". Additionally, within Greco-Roman philosophy, there were also the φαινόμενα (phainomena), "appearances" which underwent significant philosophical discussion.

In Plato's allegory, Plato asks Glaucon to imagine a prisoner in a cell in a dungeon or cave. Out of view of the prisoner is a light source. This light source illuminates an object - again out of view of the prisoner. The prisoner in the cave is only able to see the shadow cast by the object:



In terms of the Allegory, the εἶδος (eidos) is the object or vase while the μορφή (morphē) is the projected image of the object, or the shadow of the vase.

As Wikipedia notes,


  The English word "form" may be used to translate two distinct concepts that concerned Plato—the outward "form" or appearance of something, and "Form" in a new, technical nature, that never
  
  
    ...assumes a form like that of any of the things which enter into her; ... But the forms which enter into and go out of her are the likenesses of real existences modelled after their patterns in a wonderful and inexplicable manner....
  
  
  The objects that are seen, according to Plato, are not real, but literally mimic the real Forms.


So in terms of God, the form of God as it appears to us (where we are the prisoner in Plato's cave) is the μορφή (morphē). This term acknowledges that our ability to view God is imperfect, and we are not capable of truly seeing God fully and directly. In trinitarian thought, the incarnation of Jesus might be thought of in some contexts as μορφή (morphē) - as Jesus, that was how God was able to appear to us.

Conversely, God's true form could be thought of as εἶδος (eidos). It holds the concept of God's true and actual form which we lack the ability and perspective to view properly due to our humanly limitations.

John and Philippians appears to reference this as do many other passages, I am sure:


  who, though he was in the form [morphe] of God, did not count
  equality with God a thing to be grasped (Philippians 2:6 ESV). 
  
  And the Father who sent me has himself borne witness about me. His
  voice you have never heard, his form [eidos] you have never seen,
  (John 5:37 ESV)


Similarly, the Holy Spirit at Pentecost might be thought to be the φαινόμενα (phainomena) of God.

So, how do we get from form to substance?

The obvious problem for the philosopher here is that if Jesus is merely the form [morphe] of God, then he is not truly God. If Jesus is form [morphe] then he is not εἶδος (eidos). If the holy spirit is a φαινόμενα (phainomena) then she is not εἶδος (eidos).

This presents a problem for the trinitarian scripturally, with Jesus's claims to be God.

This can then easily be corrected by asking "well, what if Jesus, the Holy Spirit and God are different forms, but are all of the same substance?" What if Jesus is but is a different form [morphe] of God, but is ὁμοούσιος (Homooúsios) as God?

Conclusion

From this context, we can easily see how we arrive at the Arian Controversy. With this background in place, it is clear that when the language of Plato's Theory of Forms is used to describe Jesus as a form of God, it lends itself to the idea that Jesus is not God, but is merely a shadow or projection of God - not God himself. The language and concept of οὐσία therefore became necessary for the Trinitarians to explain both how Jesus could be both a form of God and BE God.