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One common response points out that this question makes implicit assertions that are inconsistent and self-contradictory. The phrase omnipotent being implicitly states that any phrase such as a stone too heavy for him to lift is meaningless. Thus, one solution to this type of paradox is to say that it is logically impossible for both entities to exist at the same time. So, there cannot be both an omnipotent God and an unliftable rock.
Therefore, a response is available if it is already accepted that God cannot do what is logically impossible:
This necessarily accepts the view that even an omnipotent God cannot violate the laws of logic, and indeed this whole paradox can be seen as a strong reason for such a view.
However, the answer above can be restated even if one does not already accept that God cannot do logically impossible things: one can answer that the question is literally meaningless, and therefore there is not even a logically impossible task being set.
This is a useful distinction if one wants to hold (as some, including René Descartes, have held), that God can do even some logically impossible things, such as making 2+2=4 false.
Another solution is simply that an omnipotent being can create a stone too heavy for him to lift. The being's omnipotence depends on the nonexistence of such a stone, but his omnipotence also implies that the being can give up his unlimited power if he wishes. If he were to create such a stone, then he would in effect be relinquishing his omnipotence, but no logical contradiction would arise.
Yet another solution would be to say that creating a stone too heavy for God to lift is impossible, as God can do anything. But because God can do anything he must therefore be able to create a stone so heavy that he cannot lift it: God must be able to do the impossible if he is truly omnipotent. Once the stone is created, logically, it is impossible for God to lift. However, God can do the impossible, so he can lift a stone so heavy that he cannot lift it. In the words of Harry Frankfurt "If an omnipotent being can do what is logically impossible, then he can not only create situations which he cannot handle but also, since he is not bound by the limits of consistency, he can handle situations which he cannot handle."
It could also be argued that with omnipotence comes omniscience. If God knew that he would be needing to lift the stone in the future, why would he create that stone?
The omnipotence paradox has infiltrated popular culture. In an episode of The Simpsons, Homer asks rhetorically, "Could Jesus microwave a burrito so hot that He Himself could not eat it?"
See Omnipotence for a discussion on the various positions regarding the meaning of omnipotence.
Paradoxes