The word "mum" is an alteration of momme, which was used between 1350 and 1400 in Middle English with very close to the same meaning, "be silent; do not reveal".
"Mum's the word" means to keep silent or quiet.
Mum is a Middle English word meaning 'silent', and may be derived from the mummer who acts without speaking. Note the similar English word "mime" (Old English "mīma", Latin "mimus") meaning silent actor or imitator.
Thou mightest beter meten the myst on Malverne hulles
Then geten a mom of heore mouth til moneye weore schewed!
Though thi lyppis be stokyn, yit myght thou say 'mum'.
The phrase notably appears in Shakespeare's Henry VI, Part 2, Act 1, Scene 2:
Seal up your lips and give no words but mum.