'To coin a phrase' is now rarely used with its original 'invent a new phrase' meaning but is almost always used ironically to introduce a banal or clichéd sentiment. This usage began in the mid 20th century; for example, in Francis Brett Young's novel Mr. Lucton's Freedom, 1940:
"It takes all sorts to make a world, to coin a phrase."
Coining, in the sense of creating, derives from the coining of money by stamping metal with a die. Coins - also variously spelled coynes, coigns, coignes or quoins - were the blank, usually circular, disks from which money was minted. This usage derived from an earlier 14th century meaning of coin, which meant wedge. The wedge-shaped dies which were used to stamp the blanks were called coins and the metal blanks and the subsequent 'coined' money took their name from them.
Coining later began to be associated with inventiveness in language. In the 16th century the 'coining' of words and phrases was often referred to. By that time the monetary coinage was often debased or counterfeit and the coining of words was often associated with spurious linguistic inventions; for example, in George Puttenham's The arte of English poesie, 1589:
"Young schollers not halfe well studied... will seeme to coigne fine wordes out of the Latin."
Shakespeare, the greatest coiner of them all, also referred to the coining of language in Coriolanus, 1607:
"So shall my Lungs Coine words till their decay."
Quoin has been retained as the name of the wedge-shaped keystones or corner blocks of buildings. Printers also use the term as the name for the expandable wedges that are used to hold lines of type in place in a press. This has provoked some to suggest that 'coin a phrase' derives from the process of quoining (wedging) phrases in a printing press. That is not so. 'Quoin a phrase' is recorded nowhere and 'coining' meant 'creating' from before the invention of printing in 1440. Co-incidentally, printing does provide us with a genuine derivation that links printing with linguistic banality - cliché. This derives from the French cliquer, from the clicking sound of the stamp used to make metal typefaces.
'Coin a phrase' itself arises much later than the invention of printing - the 19th century in fact. It appears to be American in origin - it certainly appears in publications there long before any can be found from any other parts of the world. The earliest use of the term that I have found is in the Wisconsin newspaper The Southport American, July 1848:
"Had we to find... a name which should at once convey the enthusiasm of our feelings towards her, we would coin a phrase combining the extreme of admiration and horror and term her the Angel of Assassination."
Sourced at Phrases.org.uk