Undoubtedly the problem of evil is the most viscerally appealing, intuitive and ancient argument against the existence of an omniperfect God. In its oldest form, the argument from the existence of evil seems to have been adequately addressed. It went something like this:
(1) If God exists, evil does not.
(2) But evil does exist.
(3) Therefore, God does not exist.
The issue with this form of the argument is that it is too easy to show that the claim made in premise 1 is too strong; any plausible theodicy shows loopholes in the idea that God and evil cannot coexist. They standardly point to some good that could only be attained by allowing some evil, such as free will (part of Augustinian theodicy) or the importance of evil in spiritual growth (Irenaean theodicy). Some evil is a consequence of free will, which is important enough to tolerate that evil. Hence, it cannot be true that the existence of God is disproved by evil, since clearly God has reason to allow some evil.
The more modern and poignant form of the argument from evil is the evidential argument where evil is presented as something which lessens the probability of the existence of God rather than outright makes it impossible. These sorts of probabilistic arguments seem too nebulous to me to consider them seriously, so instead I will refer to the logical problem of evil with a more precise first and second premise: the argument from gratuitous evil.
(1) If God exists, gratuitous evil does not.
(2) Gratuitous evil does exist.
(3) Therefore, God does not exist.
Here, gratuitous evil is defined as evil that is not necessary to achieve some greater good. Explaining evil as required for some good is obviously not going to work here, but it does at least seem like gratuitous evil exists. Natural evils such as disease or natural disasters seem, in at least some proportion, to be gratuitous. Now, I think the theist has a very good counter at this point: just because it seems unclear to us why some particular evil exists does not mean that no such reason exists. We should be sceptical of our capacity to see the reason for evil if only because from out own experience we know that hindsight has shown particular events in our own lives which we evaluated as negative at the time to have redeeming positive qualities. Whoever had a day's sickness when the World Trade Centre went down, for instance, probably was not too pleased to be sick at the time and yet was rather pleased afterwards to know that they avoided a terrorist attack. God, however, has infinite foresight, so can see all the effects of any event and would be able to evaluate whether the good outweighs the bad.
This position is commonly known as sceptical theism and I think it is essentially correct. However, it implicitly contains ramifications which are disastrous for the religious person if not adequately addressed for it says something of the nature of God: God is clearly willing to tolerate evil for the sake of good. That is the implicit assumption of sceptical theism. This means that, in some sense, God is a consequentialist with regards to God's own actions (even if not for anyone else). What if God lies to us, or at least, allows some untruth to be said by some authoritative representative? Perhaps it could be responded that God would have no reason to do so - but sceptical theism has already postulated quite reasonably that God could have reasons which are beyond our comprehension or knowledge for allowing evil, therefore, God could allow such falsehood in divine revelation.
This in turn undermines the reliability of divine revelation as a whole, for the whole of sceptical theism is the statement that the absence of evidence for a reason does not imply evidence for the absence of a reason when it comes to what God allows or not. The philosopher Stephen Law goes further and argues that sceptical theism is a downwards spiral to the pits of scepticism, since all our faculties could be faulty if God had reason to deceive us, and for all we know God does in fact have such a reason. Law claims that full bodied scepticism is the logical end of sceptical theism, and whilst I disagree, it is irrelevant to the weaker claim which is that God's reliability is undermined.
Some theists reject sceptical theism because it leads to such consequences. To reject sceptical theism, however, is to claim that we must know the reasons behind God's actions, which seems patently false in general and more certainly false from a Christian perspective: the theodicy of the book of Job, for instance, seems essentially to assert sceptical theism. The unfathomable will of God, in turn, is cited by St Paul in his epistle to the Romans as the reason that salvation comes only to those whom God has elected. So sceptical theism, from the point of view of Christianity, seems true.
What can the Christian respond? I can only see one way out: faith. Let me be clear, however, by what I mean when using the word. I do not mean to say at this moment that Christians should believe God because they should trust that God could not have reasons to lie to them. That avenue is expressly ruled out by sceptical theism as a priori. What I mean is rather more inductive: based on the relationship to God as a person, Christians should bridge the gap and trust God. This is done every day by humans everywhere; we trust people who we know could be lying because we think we know them well enough to determine that they are not, in actual fact, lying. I believe this position at least safeguards the possibility of Christians considering divine revelation to be trustworthy. Once that is accepted it becomes a self-protecting belief because the Bible makes claims about God to the effect that God cannot lie, which can be interpreted as the claim that God could not have reasons to lie in actual fact.