Curiously, there´s an interesting story on the site Drunkard
.Waiters and bartenders of the early 1900s would often keep track of running tabs by making pencil marks on the stiff cuffs of their starched white shirts, so “on the cuff” came to mean “on credit.” Considering the forgetful nature of drunks, it’s little wonder it later came to mean “on the house.”http://www.drunkard.com/02-05_on_the_cuff/
Dictionary.com goes even further with the explaininghttp://www.dictionary.com/browse/on--the--cuff
It is sometimes put as put on the cuff, meaning “extend credit to,” as in They asked to be put on the cuff until they got their monthly check
. This usage probably alludes to the practice of recording bar tabs on the bartender's cuff.
Today it´s a bit out of practice, and to give something in exchange of later payment (even with a rate of interest attached) is a bit of placing too much trust on someone else. In small towns and villages chances are, but today plastic money/cash, etc, is way more favoured.
Not to make confusion with "off the cuff" (with no or little preparation, a speech, etc. Impromptu).