Let us first see Isaiah 9:6-7 as a whole:
For unto us a Child is born, unto us a Son is given; and the government will be upon His shoulder. And His name will be called Wonderful, Counselor, Mighty God, Everlasting Father, Prince of Peace. Of the increase of His government and peace there will be no end, upon the throne of David and over His kingdom, to order it and establish it with judgment and justice from that time forward, even forever. The zeal of the LORD of hosts will perform this.
This is undoubtedly a prophecy of Jesus Christ of Nazareth. He is the promised Son of God, who fulfills all these titles and offices. Among these is the name and title "Everlasting Father." How could Christ, the Son of God, be called the "Everlasting Father"? Did He not say He came to reveal the Father (Luke 10:22; John 1:18)?
The answer lies in the answer to the question, "Who is the God of the Old Testament?" Many scriptures establish that Jesus Christ created all things (John 1:3; Colossians 1:16; Hebrews 2:10; etc.). I Corinthians 10:4 states unequivocally that the One called "the Rock" in the Old Testament is Christ (Deuteronomy 32:3-4; II Samuel 22:47; Psalm 18:2; etc.). The one called Yahweh (or YHWH) is also our Savior and King Jesus Christ!
Now notice Deuteronomy 32:6: "Do you thus deal with the LORD, O foolish and unwise people [the Israelites]? Is He not your Father, who bought you? Has He not made you and established you"? The God of the Old Testament fulfilled many roles toward Israel, and one of them is as their Father. He claims fatherhood under three bases: 1) He redeemed them—bought them—from slavery in Egypt. 2) He created them. 3) He founded them as a nation. He has every right to the title!
Finally, notice His own words in Jeremiah 31:9, speaking of the still-future time when Israel will repent: "They shall come with weeping, and with supplications I will lead them. I will cause them to walk by the rivers of waters, in a straight way in which they shall not stumble; for I am a Father to Israel, and Ephraim is My firstborn." This shows that He will be their Father when He returns to this earth to set up His world-ruling government, and this special relationship to Israel will last throughout all eternity. He is indeed "the Everlasting Father"!