In the new issue of New York Review of Science Fiction (#350, p. 1, 4-5), Michael Swanwick writes about R.A. Lafferty. He wants to argue that Lafferty "was not the solitary, literarily isolated genius, completely untouched by the work of others" that he's taken for.
He makes the point in detail by taking a Lafferty story, "Seven-Day Terror," describing it as an homage, a rewrite in an entirely different tone and style, of an obscure earlier story by Will F. Jenkins (better known in SF by his pen-name Murray Leinster, but this story, though fantasy, appeared in a non-genre magazine, in which venues Jenkins used his real name).
From my perspective of having read a dozen mutually-exclusive "discoveries" by wanna-be scholars of the one book that Tolkien must have used as the exact template for writing The Lord of the Rings, it seems to me that the shared idea in "Seven-Day Terror" - child discovers way to make objects actually disappear - is not one that a sufficiently imaginative author, which Lafferty certainly was, couldn't have come up with independently. The details of the two storylines, not to mention the tone and style, are too different to force on me the conclusion that it must be a deliberate homage. Though it might have been one.
The comparison to Tolkien comes in directly at the end of the article when Swanwick writes, "Asked for his influence on Tolkien, C.S. Lewis famously replied, 'You might as well try to influence a bandersnatch.' This was mythmaking of the first water. But it was not true of Tolkien, nor was it of Lafferty."
Yes, it's mythmaking, but that's because, by invariably taking this phrase out of context, people universally read Lewis as saying something he did not mean to say. (In this it resembles the infamous Problem of Susan, where Lewis says straightforwardly what he means, but readers take it as code for something else.) In this context, it sounds as if Lewis is saying Tolkien, like the popular image of Lafferty, had no influences, no precursors, no models. And of course that's not true, but that's not what Lewis wrote. Here's the quote in context, from a letter of May 15, 1959, to Charles Moorman, a scholar who was seeking mutual influences among the Inklings, a group of friends including Lewis, Tolkien, and Charles Williams who read their manuscripts aloud to each other:
"I don't think your project at all presumptuous, but I do think you may be chasing after a fox that isn't there. Charles Williams certainly influenced me, and I perhaps influenced him. But after that I think you would draw a blank. No one ever influenced Tolkien - you might as well try to influence a bandersnatch. We listened to his work, but could affect it only by encouragement. He has only two reactions to criticism: either he begins the whole work over again from the beginning or else takes no notice at all."
So Lewis is not talking about inspirations, the kind of influence that gets an author writing in the first place, the kind Swanwick says he's found for Lafferty. Lewis is talking about how Tolkien responds to criticism of what he's already written. Swanwick is correct that Tolkien was not entirely unresponsive to criticism - see his revision of lines in The Lay of Leithian after Lewis wrote a detailed critique, and there are other examples - but in general Tolkien did as Lewis says: he frequently redrafted (and he was a prolific redrafter) from scratch where other writers would edit what they'd already written, and when he did edit, it was more at his own initiative than from others' comments.
Where Lewis wrote in response to ideas and inspirations derived from his own reading, and this is easily traced in the finished products, Tolkien was a more hermetic creative artist whose inspirations were buried in what he called "the leaf-mold of the mind." Once he drew them out, he went his own way, and didn't treat readings at the Inklings as a writers' workshop session. (Though, of course, as Diana Glyer points out, in another sense of the word "influence," the simple encouragement that Lewis points to was enormously influential: without it, it's unlikely Tolkien would have completed either The Hobbit or The Lord of the Rings at all, and then where would we be?)