This is one of the many cases in Cassidy’s book where he ignores the correct and straightforward explanation in favour of a creaky and unconvincing origin of his own invention. As he says in the book:
But if a button is … ringing (roinn, pron. ring, to deal) in a crooked deck, every Punter is a loser. (Page 52)
In other words, Cassidy is claiming that ringing, a slang word for substitution, is from the Irish word roinn, the basic meaning of which is divide. Why a word meaning divide or deal would acquire the meaning of substitute is not explained, but then Cassidy didn’t put this one in the glossary, so presumably he was well aware that it was bullshit.
In reality, the term ringing dates back to the early nineteenth century as an expression for substitution, probably from the bell-ringing phrase ‘to ring the changes’. Then in the late nineteenth century, we get the expression a dead ringer, meaning a horse which resembles another horse and is substituted for it to banjax the gambling odds.
Cassidy’s claim is simply nonsense, like nearly everything in How The Irish Invented Slang. Incidentally, there is an even sillier explanation doing the rounds for dead ringer, that it refers to people putting telephones into graves in case they were buried alive. This just goes to show that people are absolute suckers for fake etymology.