How does Judaism deal with paradoxes relating to the omnipotence of God?
For example: Can God create a rock that he himself cannot lift?
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Questions relating to God's omnipotence were discussed at length by the Rishonim in the Middle Ages in such great works as Saadia Gaon's "Emunos V'deos", Rambam's "Moreh Nevuchim", Ralbag's "Milchemes Hashem", and others. The consensus among them (in opposition to the authorities cited in @HodofHod's answer) is that God cannot violate the rules of logic. This prevents such paradoxes such as the one you mentioned, as well as other like "Can God assume physical form?", "Can God create another God?", etc.
Why does this not undermine God's omnipotence? Because saying that God cannot do X, where X is a logically impossible task, does not limit God's power any more than saying that a person cannot drink a Wednesday (h/t @DoubleAA) limits that person's power. The verb "drink" simply does not make sense with "Wednesday" as an object. Similarly, the verb "do" does not work with "X" as an object, if X is logically impossible. Here's how Saadia Gaon says it (Emunos V'deos, Ch. 2):
ואין היא משבחתו ומפארתו אלא בתבונה ומישרים, לא בגוזמא והבל כמי שעושה החמשה יותר מן העשרה מבלי הוסיף עליהם, שמכניס תבל בחללה של טבעת מבלי צמצם האחת והרחב השניה, שמחזיר אמש לקדמותו, שכל אלה הבל הם. אפשר ישאלונו כמה מן המינים באותו ענין. אנו משיבים להם הוא יכול לכל דבר ואלו מה ששאלו, לא דבר הוא, הבל הוא וההבל אינו ולא כלום
His praises and His glorifications are only with understandable and upright things, not with exaggeration and absurdity such as "He who can make five more than ten without adding to it", or "He who can fit the entire world into the hole of a ring without shrinking the former or enlarging the latter", or "He who can make the past into the future", for all these things are absurdities. Perhaps several heretics will ask us about this. We will respond to them "He has the ability to do all things, but that which you mentioned - that is not a thing; it is an absurdity, and an absurdity is not anything."
According to some Jewish authorities, especially Kabbalists and Chassidic Rebbes, G-d does not have to obey the rules of logic, since they are just another creation of His. As I wrote to a similar question:
As the Creator of all things, including, but not limited to, time, the "laws" of physics, logic, and existence itself, G-d is not bound by any of them. Can G-d create a rock that He is unable to pick up? Yes, and He can also pick it up. How? Why? Because logic does not apply to G-d. Go argue with that (you can't; argument requires logic :-D).
From Chabad.org's "Can G‑d Create a Rock That's Too Heavy for Him to Lift?"
I've provided a long-winded answer for the inquisitive mind. Sometimes, however, it's not a philosopher asking the question, it's just some smart-aleck. But the smart-aleck also deserves an answer. So you can simply say, "Sure G‑d can create a rock so heavy that even He cannot lift it. G‑d can do anything. And He could even lift that rock that He cannot lift as well."
That'll send 'em flying. And it's not untrue. Because it's simply saying that G‑d does not fit into any of our standard ways of thinking. G‑d is not a thing—He is the source of all things. The tools of measurement of things simply do not apply to Him.
I heard someone once say that you can begin to understand God only when you acknowledge that you can never fully understand God.
Part of faith in God is understanding that we are finite and unable to ever understand the infinity of God.
So I guess the answer is that God is all powerful and we dont fully understand how that manifests itself in our world we can only have faith in that He is there and all-powerful.
To add a bit in the same vein as Jake's answer:
This is not a real paradox; it is simply a matter of definitions. "He who can make five more than ten without adding to it" is a great example of this; making five more than ten requires adding by the very definition of the ordinal system. A rock is by definition material, and lifting is by definition an act applicable only to the material.
Since God is omnipotent, he can lift anything that is definitionally subject to lifting. Therefore, to put it clumsily, "God cannot create such a rock" because God would by definition have to make it something other than a rock in order for it to no longer be subject to lifting.
I don't have any source on this but my understanding has always been that these paradoxes stem from human perceptions and interaction with the world. The hardest thing about understanding the nature of god is that it is, by definition, beyond understanding. We use a term like "omnipotence" and impose on it our version of what that means and what paradoxical limitations could be applied to it. This simply points out our inability to grasp what god is.
When people ask me that particular paradox I simply say "Yes. And then he lifts it."
In the back of the first volume of Chassidut Me'vueret on the Holidays, there is a discussion similar to this. The question asked there is, if, as the Maharal of Prague says (In his sefer Gevurot Hashem), G-d is simply whole, without any parts or definable qualities, how can we give G-d descriptions, like Merciful, etc.? How can we say the Sefirot (at least in the world of Atzilut), which are specific attributes, are an extension of G-d's light?
The answer given is based on the statement in the Avodat HaKodesh: "Just as G-d has the power of the infinite, He also has the power of the finite. For if we were to say that G-d was only infinite, and not finite, we would be limiting G-d's completion."
In other words, to say G-d can't limit himself is itself limiting G-d.
It then goes on to explain that when we say G-d is infinite, we're saying that G-d has all the advantages of everything without the deficiencies (in Hebrew this is expresses as Shlemut HaKol). Everything in the world has an advantages and corresponding deficiencies. For example, a big car might be safer in accidents, but it takes more gas and needs bigger parking spaces.
G-d has the upside of everything without the disadvantages. Going back to the Avodat Hakodesh, G-d is infinite. If you tell me that there is a deficiency in being infinite (i.e. not being able to be finite when it is advantageous to do so), I will tell you that G-d can make himself finite too, when he needs to. That finiteness is not a limitation, but an expression of his infiniteness and completion.
Here is a chapter of Heaven on Earth, by Rabbi Faitel Levin which explains this all a lot better and more thoroughly than I just did.
So, to say G-d cannot create a rock so big he can't lift doesn't take away from G-d's completion, it doesn't mean G-d is deficient. Not being able to lift a rock is a deficiency, and G-d has none.
i'm surprised that none of the answers have addressed the fact that God himself creates and abides by these contradictions. Things like God's omniscience and human free will or the kabbalistic concept of tzimtzum indicate that there are already plenty of "immovable rocks". from these examples it would seem clear that God puts limits on himself and, from our human perspective, abides by those limits. How this can be reconciled with our definition of omnipotence and "God" seems to take for granted that we understand completely what those latter two terms mean, which i don't believe is the case. If we admittedly do not know everything there is to know about God is it not the height of hubris to believe that we have "caught" him in a contradiction?