It's a bit unfair to ask me to define "natural" for you, and for science, since I reject the natural-supernatural division as intellectually incoherent, or like C S Lewis as a purely emotional distinction which, in my view, has no place in a rigorous pursuit like science. Nevertheless, there's something that draws the boundaries round science, so if we insist on the term "natural", how can we define it?
I'd have to go with Bishop Butler, I think, who wrote:
“The only distinct meaning of the word ‘natural’ is stated, fixed or settled; since what is natural as much requires and presupposes an intelligent mind to render it so, i.e., to effect it continually or at stated times, as what is supernatural or miraculous does to effect it for once.”
So my definition of natural would be couched in some terms like "lawlike patterns in the world". Thus, as pithily as I can:
Nature is that in the world which is regular or repeatable. This is the proper study of science.
I'll proceed to justify that, in the light of Joshua Swamidass's careful work on the epistemological limitations of science, to try and make it useful.
The pursuit of science has, I think, two aspects. The first is the organised observation of phenomena and the collection of data. The second, more important part is the recognition of "stated, fixed or settled patterns" in those phenomena and the building of models and theories upon them.
These patterns include both those governing individual events, such as Newton's laws, and far more commonly statistical patterns of events that, individually, are unknowable for practical reasons (such as the chaotic interactions of every molecule in a gas or the circumstances of individual genetic mutations) or for theoretical reasons (the physical indeterminacy of individual quantum events, according to Bell's theorem). These patterns constitute the only field in which science proper has a role, but it's a big role, and universally acceptable.
For a start, it has the benefit of applying to both natural and human sciences. To the extent that individual people's activities exhibit general patterns, one can make statistical predictions, whilst recognising that perhaps only unknowable individual choices make up those patterns. It could even cover, say, a statistical study of miracle claims without necessarily negating their miraculous nature, so that one could do legitimate science on what kind of diseases are commonly, or rarely, presented as cured. The sole criterion is repeatable, lawlike patterns.
Secondly, all shades of theological opinion can subscribe to it, because the recognition of patterns makes no metaphysical claims about causation. To the metaphysical naturalist, the deduced laws are primary; to the deist they are set in place by God; the theist may openly hold to divine concurrence, and even the occasionalist who believes laws are just God's habits of action may freely use the same concept, because the patterns are the subject matter, not metaphysical inferences beyond that.
But all sides will need to understand that, "beyond patterns" means, in fact, beyond nature as we have defined it and so beyond (current) science. To make a claim for "natural causes" for individual mutations, simply because there is a loose statistical pattern (as per Steve's useful post, which was essentially what I meant by mine!) is not a scientific claim, although digging down and discovering repeatable causes extends the science, of course.
But even then, to the extent that one still has to manipulate the data statistically, even such real causes cannot be assumed to be the sole ones, for you cannot keep watch to see if angels, or some more prosaic cause, is responsible for some of them.
And so the word "natural" is legitimately applied only to the realm of patterns and laws. Least of all can one invoke randomness - which only means "ignorance of causation" - as a "natural cause". It is neither a true efficient cause anyway, nor a lawlike pattern (unlike the macro-patterns that may emerge from multiple events of such unknown causes).
What this all means is that the term "methodological naturalism" will remain valid, but will have nothing whatsoever to do with the intellectually untenable "natural/supernatural" divide, but instead refers to the "natural/unrepeatable" divide. The more humbling side is that science will lose its pretence to be able to explain the whole of reality, but only by a realistic acknowledgement of its methodological limitations.
This will divide science sharply from "metaphysical naturalism", which is inextricably wedded to the "natural/supernatural" divide and claims that only "the natural" exists - clearly they could live neither with the idea that only the repeatable exists, or that the world is bigger than their system. Metaphysical naturalism may go and define itself, if it can, with more or less incoherence.