Lewis himself said "no one influenced Tolkien. He was as difficult to influence as a Bandersntch." This is often quoted as evidence of no influence, and certainly the opinion of an informed observer on the scene shouldn't be discarded. But the quote itself goes on, and Lewis notes "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."
Getting someone to start an entire work over from the beginning is influence, I would say.
Those trained to look for literary influence by pulling examples from the texts and themes will always look there first, and sometimes only. The attack of the Ents owes something to Birnham Wood coming to Dunsinane in "Macbeth." The names of the dwarves come from Snorri Sturlusen's Elder Edda. George MacDonald hearts elves. Cervantes also mentions potatoes.
How about if you influence the general subject matter - original mythology; its flavor - northern European; its form - prose rather than epic poetry; and its purpose - expressing Christianity through pagan myth. Throw in that you debate the foundational theology and theory of art with the author weekly for decades from a perspective of large agreement in a society in which there is little.
Oh yeah - also be the person most responsible for encouraging the work to completion and provide the basis for a significant character. I would wonder how one would be more of an influence than that. Authors include in their dedications their gratitude for the sacrifices and encouragement that others - usually spouses - have provided. We should take them at their word on this. You can't write an English 403 essay, much less a PhD thesis, on the influence a husband has on an author. Because that particular influence is of no use to the commenter and reviewer, it is regarded of no importance. Balderdash.
While Tolkien believed that Lewis should write less, and more carefully, Lewis believed that Tolkien should write more, and less obsessively. Fans of Tolkien's fiction would now disagree with this, but for Tolkien's professional work, it is almost certainly true. His 1936 essay "Beowulf: The Monsters And The Critics" changed criticism of that work for decades, perhaps forever. His translations of Sir Gawaine and the Green Knight, The Pearl, and Sir Orfeo were reissued 50 years after completion. (Translations are seldom reissued more than an edition or two.) He should have written more, Lewis knew it, and despaired of the scattered writing habits of his friend which brought little to completion. Tolkien had a wife and four children and was devoted to his students, it is true, and this left him little time to match the output of Jack Lewis. But what time he did have was spent much in creating the languages and histories of Middle-Earth, not for the few years of hobbit adventures, but for thousands of years, whole ages of history. While this gave the peculiar strength and foundation to LOTR that made it one of the most important books of the 20th C, it also imperiled the tale of the One Rig being told in full at all.
Lewis might marvel at being shown the three languages of Middle-Earth and say that Professor Tolkien must be the cleverest man in Oxford (and sincerely mean it), but who would have cared if there had been no burglars and dragons to draw us in? It would be a quaint legend of the Oxford English schools - a don from Merton once made up whole languages and a mythology to go with them. You can still find them over at the Bodleian. No Lewis, no Lord of the Rings.
I don't think the fantasy genre would have taken off without Tolkien. Even now, Eddings, Garner, and William Morris have few readers. None, perhaps who did not hear about them through the introduction via Tolkien's fantasy. There would have been Lewis, MacDonald, and...and what? Some Arthurian things might have pulled it through. I doubt it. No Dungeons and Dragons. Reduce the membership in the Society For Creative Anachronism by 90%. Computer games - completely different. Would the Star Wars movies have had such a ready audience? Perhaps. But no Harry Potter. The entertainment culture for those under 40 (no, 50. No, 60) would be unrecognizable.