One of the most common is to claim that causality is a physical phenomenon; it describes what takes place within the universe. You can't apply it to the beginning of the physical universe. The idea here is that causality is a posteriori like the laws of physics or chemistry, not a priori like the laws of logic. As such, it only describes the conditions inside the universe and can't be applied to the beginning of the universe itself. This is the tack taken by some illustrious philosophers, such as Adolf Grünbaum and Quentin Smith
It's certainly true that causality is not a priori in the same way the laws of logic are. We simply can't imagine the law of non-contradiction failing to hold, but we can imagine causality failing to hold -- that is, we can imagine (form a mental picture of) something popping into existence without a cause. But it's incorrect to say that we discover causality the same way we discover the laws of physics, i.e. through observation. Causality is derived from our basic intuition that something does not come from nothing (ex nihilo nihil fit: out of nothing, nothing comes). To limit this intuition to physical processes would be a case of special pleading; there's no reason why it wouldn't apply to the beginning of the universe. Causality is not a physical principle, it's a metaphysical principle.
Perhaps one could suggest that once we have the principle of causality via intuition, we can then establish it via observation and continue to believe it based on the latter. But it's not clear to me how causality could be falsified, or what would count as observation of causality not holding. At best you could say that you didn't observe a cause of an effect, but everyone would infer that the effect does in fact have a cause and we just didn't observe it. It's not like you could set up a scientific experiment to observe the absence of causality, since if the conditions you set up are sufficient to bring about the effect, then obviously the former caused the latter. As such, I think William Lane Craig's argument that causality has never been falsified is an empty claim. There are plenty of times where we observe an effect without a cause, but no amount of such experiences will ever convince a sane person that the effects didn't have a cause, merely that the causes weren't observed.
Or, perhaps one could simply deny the intuition. There are problems with this though: for one thing, science presupposes causality. If causality goes out the window, science goes with it. This is not only absurd and unacceptable, it's a conclusion I doubt nontheists would be willing to accept, since they (mistakenly) think science is on their side. For another thing, while causality is not a priori in the same way that the laws of logic are, it is still a precondition of thought. If causality did not hold, then there would not be an appropriate connection between our beliefs and their objects, such that we could never know if any of them are true. So it's not merely scientific knowledge that would be endangered; if we deny causality, then the possibility of any knowledge becomes impossible. So it's not like this intuition is just some random assertion.
But doesn't quantum physics posit virtual particles coming into existence without causes? This is a misunderstanding. As Craig writes,
... virtual particles do not literally come into existence spontaneously out of nothing. Rather the energy locked up in a vacuum fluctuates spontaneously in such a way as to convert into evanescent particles that return almost immediately to the vacuum. ... The microstructure of the quantum vacuum is a sea of continually forming and dissolving particles which borrow energy from the vacuum for their brief existence. A quantum vacuum is thus far from nothing, and vacuum fluctuations do not constitute an exception to the principle that whatever begins to exist has a cause.
Another suggestion might be that Hume denied causality. But ignoring the fact that Hume was not inerrant, this is another misunderstanding. Hume argued that just because we've observed a particular cause producing a particular effect in the past, we cannot know that the cause will produce the same effect. In other words, he argued that we can't infer an effect from a cause. Those who deny causality applies to the creation of the universe are claiming that we can't infer a cause from an effect -- that just because we observe that an effect has taken place, we can't claim that it was caused. This is radically different from what Hume was claiming, and Hume explicitly repudiates such an idea as absurd.
A final claim might be to suggest that applying causality to the Big Bang is just as problematic for the traditional theistic doctrine of creation. The doctrine, after all, is called creation ex nihilo (out of nothing) and the intuition is that ex nihilo nihil fit (out of nothing, nothing comes). But again, this is a misunderstanding. Creation ex nihilo is the claim that the universe didn't have a material cause -- that it wasn't constructed out of some pre-existent "stuff". This is certainly a radical claim and we should recognize it as such. But it doesn't deny that the universe has an efficient cause -- some entity or agent that brings about the effect -- since the claim is that God is the efficient cause of the universe. Those who deny that causality would apply to the beginning of the universe, however, are claiming that the universe had neither a material cause nor an efficient cause. So I simply put it to you, which of these two explanations is more plausible: that the universe's beginning has an efficient cause but no material cause, or that it has neither?
Now it's all well and good to say that applying causality to the beginning of the universe creates some philosophical issues, but the alternative is that it just popped into existence without any cause whatsoever. That people who portray themselves as skeptics would be willing to accept this shows that their skepticism is absurdly selective. If this is the the only way to avoid believing in God then there's just no contest.
(cross-posted at Quodlibeta)