talk the hind leg off a donkey

[Bobjex]

Okay, so I know what this means, but where did it come from? Some kind of old horse doctor, who could persuade his animals to sit down? What?

[heathfield]

Unknown origin, as are many such sayings. (The Army, Navy, and RAF have always been a very fertile breeding ground for this sort of thing, and a really good quip can rapidly spread through the services and end up in civvy street).

[jakeby]

This was discussed a while ago. No definite origin though

www.theanswerbank.co.uk/Article1100.html

[Quizmonster]

The earliest recorded use - in 1808 - of the idea mentioned "talking a horse's hind leg off", but the quote it appeared in 'way back then referred to the saying as an "old vulgar hyperbole" (exaggeration). Obviously, then, it had been around for quite some time prior to that.
I'd guess that it was based on the fact that a horse has rather long legs and anyone who can talk the hind leg off such a creature is obviously one who can go on speaking for a long time.
In other words, it may well be similar to the way we use 'donkey's years' to mean a long time, based on the original form of the saying which was 'donkey's ears'...ie something long, too.