A byword is an adage or a catchphrase — it's a motto that captures some important principle or meaningful idea. Your family's byword might be "Be kind whenever possible. It is always possible."
Like a proverb or an aphorism, a byword condenses an idea that resonates with many people into a short, memorable phrase. If you grandmother is famous for saying, "You catch more flies with honey than with vinegar," you can describe the saying as her favorite byword. In Old English, the word was biword, "proverb," or "household word," a translation of the Latin proverbium.