Well, if we consider Beetlejuice's character, he is already a rather otherworldly being. In a fanfic about Lydia's death, he might react in unexpected ways. Maybe he tries to bring her back, or he has to deal with his own feelings of loss in a way that is very different from how a normal person would. And for Lydia, her death could be a turning point in their relationship, leading to a whole new set of adventures or interactions in the afterlife.
One possible plotline could be that Lydia dies unexpectedly, and Beetlejuice, who has grown attached to her, goes on a quest to bring her back. He might have to face various challenges in the afterlife or deal with other supernatural beings who oppose his plan.