According to Protestant trinitarians, what is the significance of the father-son relationship between God and Christ? What is the significance of the father-son relationship between God and Christ in the Trinity? Non-trinitarians believe that Christ is literally the son of God, but that doesn't quite work with the Trinity. Why is Christ called the Son if he isn't God's son but is God himself?

The doctrinal and biblical basis for the Trinity has been discussed here and here. I'm not looking for biblical support of the Trinity itself. This question carries with it the basic premise that the doctrine of the Trinity is true.
I think the logical place is to start with John 8:42, and I picked this translation: King James Bible (Cambridge Ed.), because they use the term proceeded.  In others Jesus stated that he was sent.


  Jesus said unto them, If God were your Father, ye would love me: for I
  proceeded forth and came from God; neither came I of myself, but he
  sent me.


So, Jesus states that he proceeded from God, which is why God is referred to as the Father, as he is the principle (the first one) and the Holy Spirit and the Son proceeded from him, in eternity, not proceeded in terms of time, since all three exist before and outside of time.

So, Thomas Aquinas covers this well, in these links: Procession of the divine persons.

How the three relate to each other: http: The divine relations.

And this one actually covers the Father being called the Father and the Son also: The person of the Father.

We also have the Lord's Prayer where Jesus refers to God as "Our Father ...", and in various places He refers to God as His Father.

We also have Psalm 88:27 - 


  He shall cry out to me: Thou art my Father.


I am not certain if you actually want to ask about the relationship between The Father and The Son, but Aquinas dealt with that in: Whether the word "person" signifies relation?

In Article 1 in this part: (Whether relation is the same as person) , Aquinas deals with whether relation is the same as person.

I can try to sum up that, in that the three persons of God are in substance God, their relationship is only in how the proceeded from the Father, but each is still fully God, with all the properties of God, so one is not inferior to any other.