It’s a festive two-for-the-price-of-one on the origin of sayings today! (Although I must point out in advance that I’m not sure whether either term is used outside the UK.)
The first one is fairly straightforward. When I was a kid and people were imprisoned, my dad often used to say they were “in the clink” and I became so used to the term I never really gave any thought to it. (I hasten to say that I’m referring to media reports – we didn’t actually know lots of people who were sent to prison…) Well, it goes back to the days of yore
and to a secure establishment in London whose name is said to be derived from the sound of blacksmiths’ hammers making chains for captives, or possibly the sound the chains themselves made when once worn: the Clink Prison.
If you were sent to the Clink for debt, there was a separate entrance. This was on Stoney Street, hence the term (though now used largely for someone short of money rather than actually in debt): Stoney Broke – more commonly shortened to “broke” these days.