Atheists make weak arguments too. This is one of them. While it may seem bullet-proof at the first glance, let's see why it's not. So, if you use it, you will most likely lose to any non-trivial opponent. Let's get into it.
The problem with the argument is that theists can relatively easily get out of it. They can simply claim that omnipotence does not mean that a being can do EVERYTHING. Omnipotence means that a being can do anything which is logically possible. For example, it is logically NOT POSSIBLE to create a round square. It is not logically possible to make 2+2=5. It is not logically possible to have a married bachelor. Omnipotence simply means that a being can do everything that is logically possible, so, therefore, the argument does not have much sense - argument involves something not logically possible to do. It is not logically possible to make something which cannot be lifted by God.
NOTE that as a matter of formulation, you could put this proposition logically:
Let X = "to make something that its maker cannot lift."
- Wikipedia: Omnipotence paradox
But this is also not very satisfactory, because by definition God is omnipotent, so the contradiction still applies. God can lift anything which is logically possible to be lifted. So it is not logically possible to make something which is too heavy to be lifted by God.
But, there is more. "It is not logically possible to pick up a stone which an omnipotent being has made unliftable"1 So, even if he can create the stone - it is logically impossible to lift it, because it was made by an omnipotent being who decided that it's impossible to lift it, even by an omnipotent being. Since lifting it is logically impossible, it does not affect the omnipotence.
So, it seems that a theist can say either:
1. God can't make the stone - it is logically impossible to make such a stone
2. God can make the stone, but it is impossible to lift it since an omnipotent being has made the stone unliftable. But since God's omnipotence only means that he must be able to do what is logically possible, not lifting a stone does not disqualify him from omnipotence
God cannot perform logical absurdities; he cannot, for instance, make 1+1=3. Likewise, God cannot make a being greater than himself because he is, by definition, the greatest possible being.
Next, the theist can even claim that God can indeed do it - create a stone which he cannot lift, but as long as he doesn't create it, nothing happens. He stays omnipotent. Of course, this sounds rather weak, but still, it's an avenue the theist can explore which has the potential to throw you off in a debate.
If you are really bent on trying to show that God is unlikely, much better argument would be the fact that some completely unnecessary evils are happening and that this is not compatible with God who is both omnipotent and omnibenevolent - evils which cannot be explained by simply invoking the free will - for example, lightning that sets the forest on fire and animals burn slowly to death, where even if theists can argue their way out - for example - by arguing that god wishes to teach us a lesson about suffering, or to lead us to a greater good - and atheist can always state that at least some of the suffering is completely unnecesarry - as god could have prevented it, without preventing the greater good ... etc. Long story short - there are better arguments than this "stone too heavy" argument.
That all being said, some people should be reminded that an atheist has no obligation whatsoever to show that God cannot exist. The believer should show that God does, can or must exist. Atheism simply means to find those reasons not enough.
These things can go a lot deeper, so for not we'll just leave it at that, mostly because there have been like thousands of books written about these things, so the only possible way to explore even a tiny amount of them is to stick to tiny pieces at the time. Until next time.
1 The Non-Existence of God, Everitt 2004