Stumm

Can anyone let me know the origin and meaning of the phrase "mom's the word"?

"Mum's the word" is British slang meaning "I won't say anything about the subject" or "I'll keep silent on that issue". There's another now old-fashioned British expression, using mum in the exact same way - namely, keeping mum, meaning to keep silent. This was a popular phrase during the 2nd World War, as you'll see.

Mum in this sense is apparently of Middle English origin, with speculation that it's imitative, perhaps of closing one's lips.

I've heard to keep stumm about something. Is there a relationship?

Stumm, also meaning mute or silent, is a direct import from German, probably via Yiddish.