I'm guessing it comes from the poster cmpaign during the war emphisising the need to keep information confidential - spies may be listening. One of the most famous posters had the message "Be like dad, keep mum." (I know but this was sixty years ago!) and to 'keep mum' was to stay silent. This, I assume was ebreviated to the 'Mum's the word'. If Quizzy knows better, and if ayone will he will, he'll be along shortly.
The verb 'to mum' has meant to silence or to keep silent since the 14th century. Mummers were actors in silent plays, a sort of mime-performance. As a result we now have phrases such as 'mum's the word' or 'keep mum about it' and so on. "Sh!" in other words, is what it means.
I am flattered QM, but I feel that as usual, your answer is more factually accurate and informative, but your gracious response has made my day. Have a great weekend, see you on the Site next week.