Ackshually, Big Ben can never stop ticking because it never started. Big Ben is the bell housed inside the clock tower, and bells generally don't tick.
The tower is (now) known as The Elizabeth Tower. A source.
The clock itself doesn't actually have a name that I'm aware of, other than some long-winded mouthful like "The clock mechanism inside the Elizabeth tower at the palace of Westminster, the one that features Big Ben as its main bell."