Mum's the word
Keep quiet - say nothing.
'Mum's the word' has become a popular name for baby product shops and nursery services, but the 'mum' in this phrase isn't mother. Nor has 'mum' anything to do with Egyptian mummies, despite their prolonged taciturn disposition. That 'mummy' derives from 'mum' being the name of the bitumen used for embalming.
The 'mum' of 'mum's the word' is 'mmm' - the humming sound made with a closed mouth, indicating an unwillingness or inability to speak. The word is of long standing in the language and first appeared in print in William Langland's Middle English narrative poem Piers Plowman, circa 1376:
Thou mightest beter meten the myst on Malverne hulles
Then geten a mom of heore mouth til moneye weore schewed!
That loosely translates as 'You may as well try to measure the mist on the Malvern Hills as to try and get her to speak without first offering payment'.
As old as Piers Plowman, and as central to English folklore, is the tradition of mumming. Sadly, no complete texts of the mediaeval mummers' plays have been preserved. There was never a definitive version in any case, as the acting, dancing, drinking and alms collecting that made up mumming varied from one parish to another. We can't be sure what mediaeval mumming plays were like, but a raucous mixture of pantomime, morris dancing and carol singing, played out by a group of bizarre characters in stylised fancy dress, is what has come down to us by oral tradition.
What we do know is that 'mumming', or 'miming' as it was sometimes called, derives from the word 'mum'. Early versions of mumming involved a parade of characters entering houses to dance or play games in silence, that is, 'miming'. More recently, the tradition has evolved to almost always include the character of a quack doctor, who revives the hero (usually Saint George) after his death in a fight with the Turkish Knight (boo, hiss).
Although they mummed for all they were worth, the players didn't use the phrase 'mum's the word'; that usage came later, in the 17th century. The earliest version of the phrase was 'mum is counsel', that is, 'you are advised to say nothing'. That form of the phrase was used in John Palsgrave's 1540 translation of the Latin text The Comedye of Acolastus:
I dare not to do so moche as put my hande to my mouthe, and saye mum, is counseyle.
Of course, we can't examine a Tudor phrase without Shakespeare getting in on the act, and he used 'mum' in Henry VI, Part 2, 1592:
"Seal up your lips and give no words but mum."
'Mum's the word' later became the standard way of advising a person to keep quiet and the first citation of it in print that I have found is in A Walk Around London and Westminster - The Works of Mr. Thomas Brown, 1720:
But Mum's the Word - for who would speak their Mind among Tarrs and Commissioners.