The name of the Simpsons dog is Santa's Little Helper. He is named this because whilst Homer was out of money, he had a part time job as Santa at a local department store. The dog he bet on in the races did not win him any money that he was going to buy Christmas presents with, but he took the dog home as it was losing all its races and was not wanted anymore. It turned out that the dog was the best Christmas present for all the family anyway. So as he helped Homer get out of trouble, he gave the dog the name Santa's Little Helper.