Tradition that mum’s kept in the family? (6)
I believe the answer is:
omerta
If this is the answer I cannot explain how the clue works.
(Other definitions for omerta that I've seen before include "vow of silence" , "Mafia code of silence, never inform" , "Mafia pledge of silence" , "conspiracy of silence" , "Mafia code of silence; at Rome (anag.)" .)