In doing some research about this name, I discovered that Judges 13:18 also uses the same name to describe an angel. When asked what his name is, the angel replied, "Why do you ask my name? It is beyond understanding." The footnote on that passage also says "or wonderful." So that means Jesus would be a Counselor who is beyond our understanding. Since he is fully human and fully God (Philippians 2:6-7), he understands what we've gone through and yet was without sin (Hebrews 4:16), so He knows the proper way to overcome the temptations of this world.
Can you see how Truth is important in this context? As a counselor myself, part of my job is to help figure out with my clients what the problem is and then how to adequately resolve it. In essence, I am a truth-seeker and leading people towards a truth. In the same way (only perfected and full of grace and truth - John 1:14), Jesus leads people to the Truth - Himself (John 14:6).
So not only does Jesus understand our struggles, but He is perfectly able to direct us to the solution.
But let's go deeper. As a Wonderful Counselor, Jesus does what I do but much better. Being fully God, no one gives Him counsel (Romans 11:33-36) and He understands the human heart (John 2:24-25). Knowing the true diagnosis - an eternal problem (as opposed to my earthly knowledge and understanding) - Jesus is able to then offer the proper prescription.
Here's an example. In Matthew 19:16-22, a rich young man comes to Jesus for counsel. His first question illuminates his limited understanding, and Jesus' reply shows His wisdom beyond understanding. It's clear here that Jesus knew the answer to the man's question, and it's also clear that the man did not like the answer Jesus gave (since he had great wealth, Jesus' truth revealed that the man was unwilling to give it up to follow Him).
This is going to bring up a topic for a whole new post... but the fact is that the Truth hurts. Or it can, anyway. But a Wonderful Counselor, who truly loves everyone and is full of grace and truth, must be willing to speak these truths to people. Whether or not people respond to those truths is entirely up to them. Does that mean that Jesus has abandoned us or that He does not love us? No. In fact, it's quite the opposite. If you truly love someone, you will lead them to that which is best for them - even if it hurts.