Analysis:
This creationist assertion assumes two things: (I) that Big Bang theory says that the universe came from nothing; and (II) that it is impossible for something to come from nothing. Both of these assumptions are problematic.
Does Big Bang theory say that the universe came from nothing?
Big Bang theory uses Einstein's general theory of relativity to trace the history of the universe back to a moment in time when the entire universe was concentrated in a point of infinite density, called a singularity. This account of the history of the universe is simplified, because it ignores quantum mechanics: at a point in time called the Planck time (now thought to be 13.7 billion years ago), the universe is small enough to be subject to quantum mechanical effects. Exactly what impact these effects would have on the development of the pre-Plack-time universe is unknown at this point, depending on the development of a consensus theory of quantum gravity, combining general relativity with quantum mechanics.
We can thus consider from two standpoints the question of whether Big Bang theory says the universe came from nothing: from the simplified standpoint that uses general relativity alone, and from the more complete, but also murkier, standpoint that uses quantum gravity.
(i) One consequence of relativistic physics is that space and time (unified as spacetime) are themselves inseparable aspects of the universe. Therefore, if we go by general relativity alone, the origin of the universe was the origin of space and time themselves, so there cannot have been time prior to (or space outside of) the initial singularity; indeed, the very idea of "prior to" (or "outside of") the singularity makes no sense. This, in turn, shows that the vision of the universe somehow "coming from" a primordial nothingness, far from being a stipulation of Big Bang theory, is inconsistent with it.
For the universe to have "come from" nothing, it would at least have to be the case that at some point in the past, there was nothing, and then, at some later point in time, the universe suddenly existed. However, as we have seen, Big Bang theory without quantum mechanics entails that the universe existed at every moment of time there has ever been. One cannot in one breath talk about the universe existing at the first moment of time, and then, in the next breath, imply that there was a time before this first moment of time in which nothing existed.
Physicist Stephen Hawking sums all of this up when he points out that to talk about causation or creation implicitly assumes there was a time before the big bang singularity. We have known for twenty-five years that Einstein's general theory of relativity predicts that time must have had a beginning in a singularity fifteen billion years ago. (Hawking 1993:46)
(ii) What happens when one takes into account the importance of quantum mechanics in the early history of the universe? As noted above, it is difficult to say for sure, since no theory of quantum gravity has attained consensus within the scientific community. However, according to physicist Lee Smolin, there are only three possible consequences any theory of quantum gravity could have:
[A] There is still a first moment in time, even when quantum mechanics is taken into consideration.
[B] The singularity is eliminated by some quantum mechanical effect. As a result, when we run the clock back, the universe does not reach a state of infinite density. Something else happens when the universe reaches some very high density that allows time to continue indefinitely into the past.
[C] Something new and strange and quantum mechanical happens to time, which is neither possibility A or B. For example, perhaps we reach a state where it is no longer appropriate to think that reality is composed of a series of moments that follow each other in a progression, one after another. In this case, there is perhaps no singularity, but it may also not make sense to ask what happened before the universe was extremely dense. (Reformatted from Smolin 1997:82)
Possibility A gives us the same situation as that described by standard Big Bang theory: a universe which exists at every instant of time, and hence cannot have "come from" nothing. Possibility B gives us a universe extending back infinitely in time, likewise eliminating the supposed problems raised by the universe "coming from" nothing. Possibility C (which is the kind of scenario proposed in the quantum cosmological speculations of Hawking 1988) once again gives us a universe that cannot "come from" nothing, as the very notion of time-ordering ceases to have meaning in the early universe.
It thus appears that whenever a complete theory of the origin of the universe is developed by supplementing standard Big Bang theory with quantum gravity, that theory will still bypass the complaint that something cannot come from nothing.
Is it impossible for something to come from nothing?
Even were we to assume that there was in fact some time prior to the origin of the universe when there was nothing except time, it is still unclear where the problem is supposed to be. If there is absolutely nothing at all, why should one suppose there is a restriction on what can happen? There is certainly no logical contradiction in imagining there being nothing at one point in time and then something at a later point in time. It is not as though we are talking about "nothing" somehow metamorphosing into an existent something. Although the proposition that something cannot come from nothing, like the proposition that the earth is flat, has traditionally been a very popular proposition, it reflects only popular prejudice and lacks rigorous logical support. It is not that we know something can come from nothing, just that the opposite cannot be taken for granted.
(i) One argument against the idea of something coming from nothing is that we never observe such things happening. I suspect this kind of reasoning is always in the back of the mind of the average man, and explains why the idea is so counterintuitive. However, if we are talking about empty space when we talk about "nothing," then it actually is not true that we never observe things come from nothing: the quantum mechanical uncertainty principle allows for particle-antiparticle pairs to spontaneously appear out of empty space for very brief periods of time. These virtual particles (or quantum vacuum fluctuations) are ubiquitous, and create measurable effects such as the Casimir-Polder force and the Lamb shift. Some physicists have even invoked the same kind of mechanisms to generate theories of the origin of the entire universe from a background of empty spacetime (Tryon 1973).
One can, of course respond that virtual particles do not in fact appear out of nothing, because they happen in a background of spacetime in which quantum mechanics operates. While true, this response undermines the claim that we know from observation that nothing can come into existence out of nothing, since the closest thing to nothing that we are ever able to observe is empty spacetime.
(ii) Another argument against the idea of something coming from nothing is that the idea supposedly requires self-creation, which is a logical impossibility since nothing can have causal power before it exists. For instance, some people allege that to say that the universe came from nothing is to say that it created itself. But this is not so. The idea of the universe coming from nothing commits one only to the view that at one time there was nothing, and then at a later time, the universe existed. Talk of causation, much less self-causation, does not need to enter the picture at all.
(iii) Vilenkin (1982), in an extension of Tryon (1973), has proposed that quantum mechanics alone could allow for the transition of a universe with no geometry (no points) to a universe with a geometry. For the moment, I do not know what to make of this proposal, because I do not understand how one is supposed to parse the idea of a "transition" without time; however, I mention it as something for others to be aware of.
(iv) Readers are directed to Guth 1997:271-276 for a brief discussion of proposals by Tryon, Vilenkin, and Hawking.