Meaning – An exclamation of incredulity or annoyance.
There have been a few attempts to explain the origin of this odd phrase. A croze is the groove at the end of a wooden barrel that holds the end plate in place. It has been suggested that the expression was previously stow the croze, i.e. break open the barrel. I can find no supporting evidence for that idea though and have to consign it to the realms of folk-etymology. The more prosaic suggestion – that it alludes to the practise of throwing stones at crows – is much more likely.
I've found mid-20th century references from England that describe it as an Americanism and American newspaper articles that call it 'an old English phrase'. The dates of those are more or less right but not the locations – it appears to have originated in Australia. Most of the early citations in print come from there. It has a sort of Australian twang to it and is in common with several other similar phrases, all with the same meaning: starve the bardies [bardies are grubs], stiffen the crows, spare the crow.
Crows were unwelcome guests at sheep farms as, given the chance, they will kill and eat newborn lambs, so the association with annoyance isn't hard to see. The link in meaning to surprise isn't obvious, but then there's no particular reason to expect to find one. Stoning crows was a commonplace enough activity and calling it up into a phrase could have been done for no reason other than that the person who coined it just liked the sound of it. There are other expressions of surprise or annoyance like I'll go to the foot of our stairs, strike me pink, I'll be a monkey's uncle or if that don't take the rag off the bush. None of these have any sensible literal meaning and stone the crows is another to add to that list.
Those early Australian references are, from Lennie Lower's Here's Luck, 1930:
"I told Stanley that you had been thrown out and asked him to pull up, but he merely laughed and refused," he explained. "Stone the crows!" exclaimed Stanley indignantly.
Brian Penton's Landtakers, 1934 has stiffen the crows. This didn't stay the course as an everyday phrase but no doubt means the same thing.
"What I says is crows is devils." Tom pointed at the trees, where the blue-black legions sat squabbling and blinking their wicked white eyes. … "Gawd stiffen the crows," Bill commented bitterly.
The outstanding comedy writing team Ray Galton and Alan Simpson used 'stone me' in their scripts for Tony Hancock. This is a transcript taken from a 1961 episode of Hancock:
Tony: "Any room for a littl'un?". They stare at him frostily. Tony: "Cor, stone me."
Stone me has a clear ancestry back to the earlier stone the crows and was widely used, especially during Hancock's heyday. Neither phrase is now in very common use but stone me is heard more often. Galton and Simpson weren't in the business of catchphrases, although this could be called that, in the sense that it is a phrase that caught on.