The claim is that Mises predicted that US stock market crash in 1929 and (presumably) the later depression. The source of this nonsense is a story by Fritz Machlup that can be conveniently found in Skousen (2009):
“As his assistant in the university seminar which met every Wednesday afternoon, I [i.e., Fritz Machlup] usually accompanied him home. On these walks we would pass through a passage of the Kreditanstalt in Vienna [one of the largest banks in Europe]. From 1924, every Wednesday afternoon as we walked through the passage for pedestrians he said: ‘That will be a big smash.’ Mind you, this was from 1924 onwards; yet in 1931, when the crash finally came, I still held some shares of the Kreditanstalt, which of course had become completely worthless” … In the summer of 1929, Mises was offered a high position at the Kreditanstalt bank. His future wife, Margit, was ecstatic, but Lu surprised her when he decided against it. ‘Why not’ she asked. His response shocked her: ‘A great crash is coming, and I don’t want my name in any way connected with it’ … (Skousen 2009: 295–296).In the world of Austrian apologists, this prediction of the failure of one Austrian bank is transformed into the prediction of US stock market crash in 1929. Mises is alleged to have warned his future wife that “a great crash” was coming, but I have seen no evidence to suggest he was referring to America or a global depression, or anything other than the Kreditanstalt bank with that statement.
The only other evidence one can find is Mises’ introduction to the English version of his The Theory of Money and Credit published in 1934, where he claims that Austrians had “foreseen” the crisis, even though the depression had been going on for years at that point. You don’t need to be a genius to “predict” something years after it actually happens.
Skousen, M. 2009. The Making of Modern Economics: The Lives and Ideas of the Great Thinkers (2nd edn.), M.E. Sharpe, Armonk, N.Y.