There was this bat mitzvah where the girl's dog somehow got into the synagogue during the ceremony. The dog ran right up to the girl while she was giving her speech. She was so surprised but she managed to pet the dog and then said, 'Well, I guess even my dog wanted to be part of this special day!' and everyone had a good chuckle.
I heard of a bat mitzvah where the girl's dress got caught on a chair as she was about to start her Torah reading. She had to do a little dance to get free, which made everyone in the room laugh. It was a moment of pure hilarity.
One common element is technical failures. Like the lights going out suddenly during the ceremony. It makes everything seem chaotic and can really put a damper on the celebration.
One funny bar mitzvah story is about a boy who was so nervous that when he got up to read from the Torah, he accidentally read the wrong passage at first. But he quickly realized his mistake, blushed a bright red, and then started over with the correct one. Everyone had a good laugh and it became a memorable part of the celebration.