The title of this article is a conjectural. Though the topic is found within The Simpsons universe, a proper name is not available.
Hey, remember that pet gerbil we use have? He didn't run away after all!
―Bart after looking under the couch

The Simpson family's unnamed pet gerbil was a pet gerbil that was originally owned by the family.

Originally, they believed that it ran away when it went missing until when they were looking for the missing puzzle piece. Bart looked between the couch and found out that the gerbil didn't run away. Instead, it was crushed by someone sitting on the couch, which could possibly be Homer.