According to creation ex nihilo God is the origin and source of creation, creating the universe from nothing (ex nihilo). And that introduces a theodicy issue because creation ex nihilo makes God responsible for how creation turned out. If not the proximal cause of suffering then the distal cause.
Omnipotence creates a second set of issues. Even if you can get God off the hook for creating a universe that is full of suffering the doctrine of omnipotence seems to put God on the hook for not doing more to reduce or stop horrific suffering.
Now, there are no fully adequate answers to these issues, but the two ideas I've gravitated to to help with these questions--creation ex nihilo and omnipotence--are tzimtzum and cruciformity.
Tzimtzum comes from the Jewish mystical tradition and describes the idea that God creates the world by withdrawing to "make room" for creation. But because this space is a space that has been vacated by God it is characterized by formless chaos, the dark and churning "face of the deep" in the opening vision of Genesis. Another way to say this is that creation, because it is formed via tzimtzum, is characterized by God's absence.
That's Act One of the story, divine withdrawal to leave behind raw materials. Act Two is God's attempt to reinsert Himself back into creation to indwell and reign over it. In the opening scenes of Genesis this is God's Spirit hovering over the chaos, ordering the chaos to create "goodness." God doesn't say the raw material of creation is good. The chaos, the deep isn't good. What is good is ordered creation. For example, God says the light is good. Tzimtzum leaves behind a raw, chaotic material. God's Spirit indwells and orders that material to create goodness. Goodness is naming the "reign" or "kingdom" of God in and over creation.
Here is where cruciformity comes in. God reenters creation not through coercive power but through cruciformity. As N.T. Wright likes to point out, God is ruling over the world and setting the world to rights through the Lordship of Jesus. But Jesus rules and sets the world to rights through the power of the cross. Yes, God is all-powerful, but the scandal of the cross is that God's power in ruling the world appears to us as weakness, as cruciform love. God is all-powerful, but that power is always cruciform.
Why does suffering exist? I don't know.
But the best answer I have is that, because God created via tzimtzum the world is characterized by the absence of God and the raw material of creation is always tending toward chaos, disorder and dissolution. But God's Spirit is always "hovering over" creation, seeking to indwell, order and reign over creation. God is everywhere inserting light into the darkness.
And yet, the shape of this invasion and ordering is always cruciform in nature.
God reigns and rules through an invasion of love.
And if God's power ever looked otherwise, or to expect something otherwise, it wouldn't be the God revealed to us in Jesus Christ. True, in the face of suffering we might want a different sort of God, and that's fine, we can refuse to confess Jesus as Lord and say that we'd rather God set the world to rights by something other than love.
We can worship some other vision of power, some other God, Lord or Kingdom.
But if we confess Jesus as Lord we're only ever going to expect God's power among us, even in the midst of suffering, to be cruciform in nature. And that's always been the stumbling block.