This question is representative of the type of paradoxes atheists use in attempts to prove that God cannot exist. It works like this: God is supposed to be omnipotent. If he is omnipotent, then he can create a rock so big that he can't pick it up. If he cannot make a rock like this, then he is not omnipotent. If he can make a rock so big he can't pick it up, then he isn't omnipotent either. Either way demonstrates that God cannot do something. Therefore, God is not omnipotent and does not exist.
Is this logical? No. The problem is that the argument omits some crucial information and draws an inaccurate conclusion.
What the above "paradox" lacks is vital information concerning God's nature. His omnipotence is not something independent of His nature; it is part of His nature. God has a nature, and his attributes operate within that nature as does anything and everything else.
For example, I have human nature. I can run. But, I cannot outrun a lion. My nature simply does not permit it. My ability to run is connected to my nature, and I cannot violate it. So, too, with God. His omnipotence is connected to His nature since being omnipotent is part of what He is. Omnipotence, then, must be consistent with what He is and not with what He is not since His omnipotence is not an entity to itself. Therefore, God can only do those things that are consistent with his nature. He cannot lie because it is against his nature to do so. Not being able to lie does not mean He is not God or that he is not all powerful. Also, He cannot cease to be God. Since He is in all places at all times, if he stopped existing, then he wouldn't be in all places at all times. Therefore, He cannot cease to exist without violating his own nature.
The point is that God cannot do something that is a violation of his own existence and nature. Therefore, He cannot make a rock so big he can't pick it up or make something bigger than himself, etc. But, not being able to do this does not mean He is not God nor that he is not omnipotent. Omnipotence is not the ability to do anything conceivable but the ability to do anything consistent with His nature and consistent with his desire within the realm of his unlimited and universal power, which we do not possess. This does not mean He can violate his own nature. If He did something inconsistent with his nature, then he would be self-contradictory. If God were self-contradictory, he would not be true. Likewise, if He did something that violated his nature, like make a rock so big he can't pick it up, He would also not be true since that would be a self-contradiction. Since truth is not self-contradictory and neither is God and if he were not true, then he would not be God. But God is true and not self-contradictory. Therefore, God cannot do something that violates his own nature.
A rock that big is impossible
Another way to look at it is to realize that in order for God to make something so big he couldn't pick it up, He would have to make a rock bigger than himself. Since He is infinite in size, he would have to make something that would be bigger than himself. Since it is His nature to be the biggest thing in existence, because he created all things, He cannot violate his own nature by making a rock that is larger than he.
Also, since a rock, by definition, is not infinitely big, then it isn't logically possible to make a rock, something that is finite in size, be infinite in size (no longer a rock) since only God is infinite in size. At dictionary.com, a rock is defined as a "Relatively hard, naturally formed mineral or petrified matter; stone. a) A relatively small piece or fragment of such material. b) A relatively large body of such material--as a cliff or peak. c) A naturally formed aggregate of mineral matter constituting a significant part of the earth's crust." A rock, by definition, is not infinitely large. So, to say the rock must be so big that God cannot pick it up is to say that the rock is no longer a rock.
What the critics are asking is that God become self-contradictory as a proof he doesn't exist. Their assertion is illogical from the start. So what they are doing is trying to get God to be illogical. They want to use illogic instead of logic to prove God doesn't exist. It doesn't work, and the "paradox" is self-refuting and invalid.