So far we've been introduced to Jesus as "wonderful counselor" and "mighty God." Next up we learn He is called "everlasting Father." Immediately, you probably notice there are two main things that are brought up here–time and relationship.
Jesus is not just being introduced to us as a powerful authority with wisdom and a God for a time, but He is a God for forever. Second, He is being introduced to us as Father. This is an important shift in Jesus' names so far. The first two names had to do with who Jesus was as a king or ruler. Now we begin to learn how He rules, and the way He rules is relational.
He doesn't want to just be known a king in authority, He wants to be known as a Father.
Personally, I think this is where the truth of the gospel becomes incredibly compelling. There are many "gods" and religions that introduce us to a "god" with authority over man or some kind of wisdom. But Jesus longs to be in relationship with the people who He rules, He wants us to call Him Father, and to know Him as Father forever.
In His first two names we are to recognize that He is a God of authority, then we get personal and start to hear how He changes our identity with His identity–He is an everlasting Father, inviting you to be His kid, forever and ever.
The name of "everlasting Father" shows us that Jesus wants to point us to the reason He came: He wants to be your Father for eternity. As our eyes are directed to eternity in His name, we being to see that He desires to be a Father to us forever.
A God who is also Father will act as a Father; Fathers love, care, protect, provide, and desire the absolute best for their kids. Later, in Matthew 7:11 Jesus asks, "If you then, being evil, know how to give good gifts to your children, how much more will your Father who is in heaven give good things to those who ask Him!"
Focus on knowing God as your everlasting Father and meditate on how He desires to reign His kingdom through relationship.