According to the Harry Potter Wiki, the name of Argus Filch's cat was Mrs. Norris.
Mrs. Norris was a tabby cat owned by Argus Filch, the caretaker of Hogwarts School of Witchcraft and Wizardry. She was known for patrolling the halls of Hogwarts, searching for students who had broken the rules. She was very loyal to Filch and was also known for her sharp hearing and sight.