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The letters that derive from $\mu$ include the Roman M and the Cyrillic М. Hence considering that the word "mean" starts with an $m$ the choice seems relatively straightforward given an already existing tradition to use greek letters in mathematical abbrevation.

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Fisher wrote the normal density with $m$ for the mean (see section 12 of his Statistical Methods for Research Workers) until the mid-1930s when he replaced $m$ with $\mu$. The new symbol appears in The Fiducial Argument in Statistical Inference (1935) and it went into the 1936 (sixth) edition of the Statistical Methods for Research Workers.