How do language researchers find the first written mention of a word?

Solution 1:

I answered a similar question on Linguistics SE, which I will plagiarize in part here:

How etymological research is done has varied through time. In the case of the "New English Dictionary" (the first edition of the Oxford English Dictionary), work started on it in 1857. Then:

[I]n January 1859, the Society issued their 'Proposal for the publication of a New English Dictionary,' in which the characteristics of the proposed work were explained, and an appeal made to the English and American public to assist in collecting the raw materials for the work, these materials consisting of quotations illustrating the use of English words by all writers of all ages and in all senses, each quotation being made on a uniform plan on a half-sheet of notepaper, that they might in due course be arranged and classified alphabetically and by meanings. This Appeal met with a generous response: some hundreds of volunteers began to read books, make quotations, and send in their slips to 'sub-editors,' who volunteered each to take charge of a letter or part of one, and by whom the slips were in tum further arranged, classified, and to some extent used as the basis of definitions and skeleton schemes of the meanings of words in preparation for the Dictionary.
_{An Appeal to the English-Speaking and English-Reading Public to Read Books and Make Extracts for The Philological Society's New English Dictionary}

One significant contributor to the early OED worth mentioning is William Chester Minor (1834 – 1920). He was insane, but he was also good at doing etymological research. His story, graphic in some parts, can be found here:

What made him so good, so prolific, was his method: Instead of copying quotations willy-nilly, he’d flip through his library and make a word list for each individual book, indexing the location of nearly every word he saw. These catalogues effectively transformed Minor into a living, breathing search engine. He simply had to reach out to the Oxford editors and ask: So, what words do you need help with?

The "Reading Programme" is still used by the OED, although the methodology is different. The books are still read all the same but here's what happens next according to a freelance researcher for the OED:

I then consult OED Online to determine whether the word or phrase is in the Dictionary: if it is not, I submit it as a ‘not-in’, and if it is, I decide whether its form or context is important enough to warrant its submission. If it does qualify, I enter the information into tagged fields in an electronic file that has been set up in a standard format. When I have finished the reading, I submit the file to Oxford or New York, where the records are incorporated into OED‘s working database for consideration by the editors, along with thousands of paper citation slips, as they proceed through the current revision. Yes, some of my finds are still submitted as paper slips—a reminder of OED‘s long heritage—but, electronic or paper, I can hardly imagine a better job.

The quotations were collected in a machine readable format for the first time in 1989. The 1990 UK Reading Programme captured material electronically. (Note that the second edition of the Oxford English Dictionary came out in 1989.)

In addition to this, the OED now utilizes several online databases of texts, such as Early English Books Online, Eighteenth Century Collections Online, and some newspaper databases.

I have access (for now) to several of these paywalled databases through my college.

If you do your own research with databases (many people use the free Google Books), it's often easy to find antedatings for pages that haven't been updated for the third edition of the OED. Updates to the OED3 started in 2000 and continue to this day: it's a huge dictionary and updating takes time.

See also:

• OED: Researching the Language

Solution 2:

The original, most authoritative source on these in pre-digital times was the Oxford English Dictionary. And they assembled all of their references manually, although they can be said to have invented crowdsourcing, as people from all over the world.contributed.

The OED is still an amazing source. Today, though, companies like Google and efforts like Project Gutenberg are making great efforts to digitize everything we have, and then it's only a 'grep' command away. Like Tuffy said, a good deal of everything we ever wrote is lost, though, and having studied ancient Greek myself, it's tragic. We've lost countless works of some of authors we, and even everyone at the time, knew were masterpieces. Try searching for Sappho, so many fragments. Whole plays by Euripides, gone.