I do thank you, my dear friend, for your very good question. In short, this particular verse is a prophetic reference to the Christ, the Messiah, specifically, the Lord Jesus Christ. We must understand here that the Lord Jesus Christ is both true man and true God, and each of the names in verse 6 refer to the Lord Jesus, and specifically to the various characteristics by which the Christ would be known. I believe that all this is important to the explanation as to why the “Son of God” could be called “the everlasting Father.” I believe that the words of a couple of Bible scholars will be most helpful here. First, I will quote the MacDonald Bible Commentary on this verse, “Everlasting Father – or better, the Father (or “Source”) of eternity. Eternal Himself, He confers eternal life on those who believe in Him. Vine comments: “There is a twofold revelation in this: (1) He inhabits and possesses eternity (57:15); (2) He is loving, tender, compassionate, an all-wise Instructor, Trainer, and Provider.” F. B. Hole shares a similar view in his commentary on Isaiah: “He is “Mighty God.” Again, being so, when He takes flesh and blood, His name of course must be, “God with us.” Moreover, He is “Father of eternity,” as more literally the words read. Eternity has its origin in Him. The ascription of Deity to the Child born could not be more distinct.”

Now, for some further information on Isaiah 9, the chapter as a whole spells out the judgments which are upon the children of Israel due to their infidelity to the Word of God; however, in verse 6, there is a ray of light and hope. The prophesy plainly tells that the Messiah will come to the house of Israel. He will be born of a virgin in the district of Galilee, which had long been in spiritual darkness, and His personal glories are foretold in the several names listed there, each and every one of them bringing out some particular characteristic of the Christ. Immediately in Isaiah 9, the LORD focuses the attention upon the Kingdom of Israel, highlighting the northern portion of which has already gone into captivity years before, and that part which contains the region of Galilee where the Christ child will spend His early years. We see this district identified by name in Isaiah 9:1. Yet, in the midst of the hard judgment of God on the people, God has revealed His plan for deliverance by sharing a great light amid the existing spiritual darkness-a great deliverance, the Christ of God, is coming, who will one day set all things right. We know from other Scriptures that Christ Jesus will reign in righteousness (the government will be upon His shoulders). We know that Christ Jesus will, when He returns to the earth the second time in power and glory, will in fact reign upon David’s throne, healing the faithful of His people and the land (see Malachi 4:2 and Revelation 11:15).  

Now as to the deity and humanity of Christ, we read in verse 6 about the coming of the Christ child into the world, and we read there: “unto us a child is born,” this phrase expresses the humanity of Christ in that He was born of a human mother, a virgin, and thus He was fully “man.”. The second part of the expression goes on to tell us that “unto us a Son is given.” God the Father is the true and only Father of Jesus Christ, and because of this and because there is no human father, this child is true God, the only begotten “Son” of God. Thus, I believe that the full humanity and as well, the full deity of Christ is clearly brought out in this phrase. and you may note also in verse 6 that this “Son” is “given” rather than to be “born”. To me, this emphasizes that the Christ, God the Son, predated the birth of baby Jesus.  In other words, Jesus was God from the beginning, and God gave to us His Son, who existed in eternity past, long before He came into the world as a baby (John 1:1 and 14). Thus, in this short verse from Isaiah 9, we learn all about the Messiah who was to come. We learn about His manhood and also His deity, His eternal existence, the district of Galilee where Jesus would be from, and what was so significant about Galilee, and the role that Messiah is to play in the end times.  (SF)  (678.2)