Buckley's chance

Uncertain. Candidates are:

A reference to William Buckley (1780–1856), a convict who escaped in Victoria in 1803 and lived with an Australian Aboriginal tribe for 30 years (non-Aboriginal people in the bush were considered to have "no chance" of survival). This is the most popular candidate, but earliest known usages date from the 1890s, some 30 years after his death. Nonetheless, this was soon after Buckley's story was publicised in 1889 by E. W. Cole in his book Savage Life in Australia: The Story of William Buckley the Runaway Convict who Lived Thirty-two Years Among the Blacks of Australia, so this etymology remains plausible.

From Buckley's and none, presuming that expression derives from a Melbourne department store, Buckley & Nunn.

A reference to Mr Buckley of the Bombala region of southern New South Wales, who sued the government over title to land, the action seeming to have little prospect of success.

Again a reference to Mars Buckley, not in connection with Crumpton Nunn (as above, re Buckley's and none), but rather in relation to an 1893 run on banks, when Buckley ensured that the Bank of Australasia would have no chance of using his money to pay other depositors. The bank thus had "Buckley's chance" of getting his money. This etymology is arguably somewhat likely since the phrase was first cited three years after this incident.