The central tenet of theodicy is that there is a problem of evil. More technically, theodicy exists because people find a logical inconsistency between their idea of a good and powerful God and their ideal universe and they wish to remove this inconsistency.
Here is a version of the "logical problem of evil" which I copied from Wikipedia. 1. God exists. 2. God is omnipotent, omniscient, and perfectly good. 3. A perfectly good being would want to prevent all evils. 4. An omniscient being knows every way in which evils can come into existence. 5. An omnipotent being who knows every way in which an evil can come into existence has the power to prevent that evil from coming into existence. 6. A being who knows every way in which an evil can come into existence, who is able to prevent that evil from coming into existence, and who wants to do so, would prevent the existence of that evil. 7. If there exists an omnipotent, omniscient, and perfectly good being, then no evil exists. 8. Evil exists (a logical contradiction).
There is an unacknowledged assumption hidden amongst these propositions, the assumption that "I know what a good God would want to do". How do you know this? From personal experience?
Revelation, as represented by the Hebrew scripture, doesn't support the premises of this logical problem. The Bible declares that God is good and also that creation is good. If the world is good, then there really isn't a problem of evil which one would need to resolve.
Actually, Genesis leaves open the possibility that creation was good but is not so very good any more. The story of the garden (known as the story of The Fall among many theodicists) suggests that evil only appeared when humankind ate of the fruit of the tree of the knowledge of good and evil. Many say that the eating of the fruit was the Original Sin which brought evil into a world which was previously good. It isn't clear to me how evil is the consequence of this act, since it would seem that the act which brings evil into the world must itself be evil, and also the desire which led to the overt act, and so on backward into the good time in the garden, but let us leave that question to the side.
My own interpretation would be that evil was not a defined category until human beings, reaching beyond their animal nature (to become "like gods" as the story says), created for themselves the categories of "good" and "evil", "us" and "them", "lawful" and "unlawful", "my idea" and "mistaken". It is not unreasonable to suppose that the problem of evil might be a categorization error.
Taking that point of view, one can go a bit farther and claim that the problem of evil, and therefore also theodicy, is the fruit of the tree of the knowledge of good and evil. Judging by some recently popular books, we're still eating it up.
The revelation of the Christian scriptures adds another point, which is that God suffers. What is more, God chooses to suffer. I asked before, how do you know what a good and omnipotent God would want to do? According to revealed truth, a good and omnipotent God would -- and in actual reality did -- choose to suffer.
One needs to be careful when expressing the truth in this way, because it is all too easy to twist this thought into justification for making someone else suffer, which isn't the point at all. Or it can be twisted into a glorification of self-inflicted pain, medieval European self-flagellation being the type specimen for that error.
According to the Christian story, God's choices do not easily map into our human ideal of a good universe. God surprises us. We may not understand what a good and omnipotent God would want. In humility, we should assume that we do not know.