Can God create a rock so heavy that even he cannot lift it? If he can, then the rock is unliftable, God is not omnipotent. But if he cannot, then he is still not omnipotent.
I asked my 8 year old daughter a variation on this question and I am not aware she had ever been exposed to it before. I tried not to direct her answers so I could see what she thought.
Bethyada: Can God do anything?
B: Can he make a rock so large he cannot lift it?
B: Can God do anything?
B: If God cannot make a rock so large he cannot lift it explain why God can do anything.
D: Well, if it is too big he can't lift it. If he can lift it he can't make it. [showing she understands the dilemma]
B: So then God cannot do everything. Can God do anything?
B: How, what is wrong with the question?
D: It is a silly [read illogical] question.
So in essence she showed she understood the dilemma, this didn't shake her belief that God is omnipotent and she stated the problem is actually with the question.
This has been dealt with centuries ago, but as it continues to be mentioned in various forms so let us review it.
Starting with the first part of the question consider a group or a set of of big (immovable) rocks that God cannot lift. Call this set R (for rocks). Now if God could make such a rock, then set R would have at least 1 rock in it and therefore it would not be empty. However if God cannot make such a rock then set R is empty.
So R could be either empty or not empty. However it cannot be both. This is a logical impossibility.
The second half of the question states that of he cannot make it (R is empty) he is not omnipotent and if he cannot lift it (R not empty) he is still not omnipotent. The dilemma does not need to bring God into it because it has to do with a foolish definition of omnipotence that denies the law of no-contradiction, that a set be both empty and not empty at the same time.
Omnipotence means that God can do all meaningful things.
This applies to any dilemma that expects God to do the logically impossible but can be extended to any dilemma that expects God to have 2 mutually exclusive traits, either attributes that are intrinsic to his nature (eg. goodness) or something he wills to do (eg. second coming).
This does not say that God cannot have 2 attributes that are generally considered difficult to reconcile. God can be both just and merciful. He cannot be both just and unjust.
Are there any scriptures that discuss this. I think the following is relevant:
If we have died with him, we will also live with him;God is faithful and that precludes him from being faithless.
if we endure, we will also reign with him;
if we deny him, he also will deny us;
if we are faithless, he remains faithful—
for he cannot deny himself. (2 Timothy 2:11-13 ESV)
What if someone insists on a meaning of omnipotence that includes the ability to do the illogical? Other than acknowledging the person is an idiot, one could say that in that sense God is not omnipotent.
Does all this mean God is limited by logic? No. Logic is a component of God's nature. We can understand logic because God gave us logic as part of imparting his image in us; but, this is another topic.
Now if only more people could have the insight my daughter has at 8!