Back in 1994 I came to embrace what is commonly called Calvinism and is better termed "the doctrines of grace." This helped me take passages like Ephesians 1:1-14 and 2:1-10, Romans 8 and 9, the Gospel of John, and many other passages at face value. I no longer had to ignore or explain away the New Testament teaching concerning predestination, election, mankind's deadness concerning spiritual matters, etc. There was clarity.
At that time, I would have been wary of any talk of "mystery" in the debate concerning God's sovereignty and man's free? will. It would seem any ground taken away from God would have to be parceled out to man, which didn't seem right.
Eventually, however, D.A. Carson helped me see that there is indeed a mystery. It is not so much a mystery concerning who wins in the battle of God's will and man's will, the classic tension. The Bible, I think, makes the answer to that one clear. Instead, it is a mystery within God Himself: He is both sovereign (His will reigns supreme) and personal (He interacts with humans, meaningfully).
So, in the Bible we see both these realities at work: Jesus was the Lamb slain before the foundation of the world; those who nailed Him to the cross did so according to the eternal decree of God (Acts 2:23). Yet Jesus cried out in prayer in the Garden of Gethsemane, pouring out His heart while surrendering to the Father's will. No Stoicism. Or, look at the missionary passion in the book of Acts: apostles who knew God was responsible for anyone's eyes being open to the gospel (Acts 13:48) yet prayed and preached up a storm to see it happen.
God is sovereign, and God is personal.
God calls the shots, and God truly, meaningfully responds to prayer.
What is commonly called Arminianism actually seeks to alleviate the mystery. This position claims, "Here's how it works: God looks down the corridor of time and sees who will choose Him; then, He chooses them." This attempts to give God's will and man's will equal footing, but really end up selling the farm to man--at the crucial moment, God is passive. While this sounds like a theological option, I don't think this is what the Scriptures as a whole teach.
Meanwhile, Calvinists can try to alleviate mystery another way. I think I've seen this more in deeds than in words, and it's what drives those who are more free-willish in their thinking crazy. It's when Calvinists undermine prayer, preaching, or sacrificial living by an attitude that speaks louder than words, "God will do what God will do, I will go back to my books now." It's not so much about how one is saved (it's all of God!) but of the means God uses. For example, He uses the impassioned prayers of those who love their friends and desire to see them saved--in short, he uses us!
I should note that God's choosing is also mysterious. The Bible refers to God's choice being based on love, and being for His glory, but does not give us an algorithm. The biggest mystery is that God would set His love on any of us! It certainly does not permit someone dead in sin to boast in the gift of resurrection life.
The Reformed tradition calls this the mystery of providence. And it uses the term "secondary causes" to refer to those means by which God accomplishes His will, including the examples I've given here already -- prayer, preaching, and so forth.
So here is my position: There is a mystery, but it is a Calvinist mystery. And fully believing in God's sovereignty is actually necessary for there to be a mystery.