My family has one that is told around christmas time.
My uncle started doing this for us every christmas morning when we would wait for our parents and aunts and uncles to wrap the presents outside. He told us the story of the patient donkey. basically what happened was a donkey and his owner were walking in a desert on their way to an oasis. The donkey asked if they were almost there many times and each time the owner would respond with, patience, donkey, patience. He would repeat that until our parents were done with our presents and they let use out. I am not sure if that is the type of tradition you were talking of.
We also have another where on each others birthdays, no matter how early they have to wake up, we would wake them up singing happy birthday.
Another odd one, lol, is on the first day of every month the first person to say "rabbit rabbit" wins (that one is strange lol)