E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
To bend the point of a nail after it is driven home. The word is sometimes written clench, from the French clenche, the lift of a latch. (German, klinke; Dutch, klinken, to rivet.) (See page 261, col. 1, CLENCH.)
That was a clincher. That argument was not to be gainsaid; that remark drove the matter home, and fixed it as a nail in a sure place.
A lie is called a clincher from the tale about two swaggerers, one of whom said, I drove a nail right through the moon. Yes, said the other, I remember it well, for I went the other side and clinched it. The French say, Je lui ai bien rivé son clou (I have clinched his nail for him).