Took is one of the few names Tolkien claims not to have “Englished” — that is, adapted into the translation conceit by which he explained so many other names. These “one or two older names of forgotten meaning [which Tolkien was] content to anglicize in spelling” in-cluded “Took for Tûk and Boffin for Bophîn”.  In the “Nomenclature”, Tolkien echoes this: “Took. Hobbit-name of unknown origin representing actual Hobbit Tūk […]. It should thus be kept and spelt phonetically according to the LT [i.e., the Language of Translation].” 
From a story-internal point of view, this is perfectly plausible, but from the story-external vantage, why wouldn’t Tolkien just “come up with something”? A possible answer is that he was stuck with Took from The Hobbit, long before Middle-earth had come into focus and the translation conceit entered Tolkien’s mind, and he simply couldn’t think of anything. Or perhaps there was a source, but it simply wasn’t appropriate for or adaptable to The Lord of the Rings.
I’m not aware of any real source criticism on this name, not even by my friend, Mark Hooker, who has worked his way pretty systematically through the “Nomenclature”. Perhaps the claim of invention on Tolkien’s part has discouraged scholars and dictionary divers. But let’s not be discouraged!
It turns out that Took, like Boffin, Grubb, Bolger, so many others of the Shire and Bree, is a real British surname. Ernest Weekley points out that the genuine name, Tooke, derives from the Anglo-Saxon Toca.  The Anglo-Saxon name, in turn, apparently derived from an Old Norse name Tóki, but had become naturalized in the southern part of England by the 11th century.  Tom Shippey has noted the survival of the name: “As for ‘Took’, that too appears a faintly comic name in modern English (people prefer to respell it ‘Tooke’), but it is only the ordinary Northern pronunciation of the very common ‘Tuck’” .
Another possibility: it occurs to me that Tolkien might even have chosen Took as the name for his most adventurous hobbit-family in facetious reference to his own name, Tolkien, which glosses (more or less) as “foolhardy”. If so, then this would seem a perfectly appropriate choice.
So, it’s pretty clear to me that Tolkien might have resurrected the genuine English Took(e), just as he did Gamgee, Brandybuck, Bracegirdle, Hornblower, and all the rest. And/or he may have been thinking of the etymology of his own name. Is there any more to be said? Yes, just a little, and here’s where things get more interesting — but more wildly speculative. It just so happens, there was a rather well-known philologist by the name of Tooke!
John Horne Tooke (1736–1812) was a Cambridge-educated etymologist and politician. His best-known work, Επεα Πτεροεντα, or the Diversions of Purley (1786), is a collection of philological dialogues on subjects such as: “Of the Division or Distribution of Language”, “Etymology of the English Conjunctions”, “Of the Article and Interjection”, “Of Participles”, and so on. The kind of thing that was right up Tolkien’s street.
In 1805, a reviewer assessed Horne Tooke’s impact on lexicography, thus: “to him the English language owes the pristine introduction of just principles, and a most extensive, learned, and detailed application of them to the etymology of its terms. He has laid the groundwork for a good Dictionary” . But this was an early opinion. Samuel Taylor Coleridge was more critical, writing in 1830 that, although “Horne Tooke was pre-eminently a ready-witted man[, h]e had that clearness which is founded on shallow-ness. He doubted nothing; and, therefore, gave you all that he himself knew, or meant, with great completeness. […] All that is worth any thing (and that is but little) in the Diversions of Purley is contained in a short pamphlet-letter […]” .
Indeed, philology has come a long way since Horne Tooke’s days. Most of his ideas have been superseded or proven patently wrong (e.g., his etymology for “Shire”  is clearly incorrect ). He is viewed nowadays as somewhat of a crackpot.  But there can be no doubt that Diversions of Purley made quite a splash, one whose ripples were felt throughout the 19th century, inspiring both argument and imitation. It was clearly a part of the zeitgeist of the century, the lexicographical culmination of which was the launching of A New English Dictionary on Historical Principles (now known as the Oxford English Dictionary). Tolkien himself was employed by the OED in 1919–20. I do not know of any evidence that Tolkien was aware of Horne Tooke or his philological work more than a century before, but James Murray, one of the original editors of the OED, certainly was . Murray died shortly before Tolkien’s appointment to the Dictionary, but it is tantalizing to think that Tolkien might have learned of Horne Tooke during his tenure in the Old Ashmolean. I know of no reason to assume he didn’t know of him.
Assuming Tolkien learned of Horne Tooke, perhaps even read his work, is it possible the name stuck in his mind, only to reappear a decade or so later as an “Englishy” surname in his children’s book, The Hobbit? A notorious philologist named Horne Tooke is tempting quarry. Even Horne, suggesting a musical instrument, faintly recalls the hobbit names Hornblower and Bullroarer. It’s probably just coincidence, but it’s certainly not impossible that the name influenced Tolkien. After all, Tolkien made reference to lexicography elsewhere in his fiction, as in the “four wise clerks of Oxenford” in Farmer Giles of Ham.
 Tolkien, The Lord of the Rings, Appendix F, III.
 Tolkien, “Nomenclature of The Lord of the Rings”, in Hammond, Wayne G., and Christina Scull. The Lord of the Rings: A Reader’s Companion. Boston: Houghton Mifflin Harcourt, 2005. 750–782, p. 764.
 Weekley, Ernest. The Romance of Names. 3rd rev. ed. London: John Murray, 1922, p. 75.
 Smart, Veronica J. , 280. “Moneyers of the Late Anglo-Saxon Coinage: the Danish Dynasty, 1017–42.” Anglo-Saxon England 16. Cambridge: Cambridge University Press, 1987. 233–308, p. 280
 Shippey, Tom. The Road to Middle-earth: How J.R.R. Tolkien Created a New Mythology. Rev. and exp. ed. Boston: Houghton Mifflin, 2003, p. 103. I’m not sure Tom is correct that Tooke and Tuck are the same name; I’ve read contrary views. Tuck and Tucker seem to be vocational names, but I’ve seen no such theory advanced for Tooke. But I’ll keep looking.
 Quoted in Tooke, John Horne. Επεα Πτεροεντα, or, The Diversions of Purley. New ed., rev. and corrected, with notes, by Richard Taylor. London: Thomas Tegg, 1840, p. xiv.
 Coleridge, Samuel Taylor. Specimens of the Table Talk of Samuel Taylor Coleridge. Ed. Henry Nelson Coleridge. 2nd ed. London: John Murray, 1836, p. 62 (see also passim).
 Tooke, p. 424.
 See for example, Skeat, Walter W. An Etymological Dictionary of the English Language. 3rd. ed. Oxford: The Clarendon Press, 1898, p. 548.
 See Lynch, Jack. The Lexicographer’s Dilemma: The Evolution of ‘Proper’ English, from Shakespeare to South Park. New York : Walker & Co., 2009.
 See Mugglestone, Lynda. Lost for Words: The Hidden History of the Oxford English Dictionary. New Haven: Yale University Press, 2005.