THE NAME
Why on earth is this series called Fruits Basket? Well, in the series Tohru feels she doesn't belong with the Sohma's. Just like how a riceball (Also known as Onigiri in japanese) doesn't belong in a Fruits Basket.
She compares this to when she was little and her classmates would play the game called, Fruits Basket. They each would take a fruits name and who ever's name comes up gets to go and play. They would tease Tohru all the time and call her the rice ball and without thinking Tohru would sit there smiling waiting for her to be called, only to find out that a riceball doesn't belong in a fruits basket.
So she feels that way about living with the Sohma's when she hears but happily she is finally called, riceball, and is allowed to be with them, to belong.
That also explains why they make a riceball her 'sign' in the series.