I think the classical response is Leibniz’ “best of all possible worlds”-argument: out of all logically possible worlds, God chose the one with the least evil; doesn’t mean that it has to be zero, or even a particularly small amount.
For those who think that’s not enough, there’s Plantinga’s free will defense: basically, the assertion of the problem of evil is that the existence of evil is logically incompatible with the claimed attributes of God. Plantinga refutes this by exhibiting an explicit set of circumstances where, under an additional assumption (the existence and valuability of free will), these are compatible, after all, thus proving that there never was any incompatibility in the first place. Note that in order for this argument to work, there need not actually be free will!
Perhaps more explicitly, the problem of evil is the assertion that there is a contradiction between the four propositions:
- God is omnipotent.
- God is omniscient.
- God is omnibenevolent.
- There is evil in the world.
The contradiction is (apparently) straight forward: an omnibenevolent God would want to eradicate all evil, an omniscient God knows whenever there is evil, and an omnipotent God can perform the necessary actions to abolish all evil. Thus, if God is omnipotent, omniscient, and omnibenevolent, there is no evil. But there is evil, hence, God is not omni-etc.
Plantinga now includes another proposition: roughly that, all else being equal, a world in which there are free beings is more morally valuable than a world in which there aren’t. Thus, if there were free will, then the conclusion above does not follow. But then, the conclusion does not follow at all, regardless of there actually being free will, since there exists a logically consistent set of circumstances in which it is false, and thus, 1, 2, and 3 do not entail not-4.
This works just as well for the problem of natural evil, if one imagins there to be superpowerful beings (“demons”) capable of producing natural desasters, who are also more morally valuable if they are free (and thus, free to do evil). The philosophical consensus, as far as I’m aware, is that this is a completely satisfying rejoinder to the problem of evil.
Of course, I’m an atheist anyway, so I don’t have any problem of evil in the first place. 