LORELAI: Look, I didn’t come here to make up, or to try to get you to forgive me, or talk. I wouldn’t even have come here at all but I had a really crappy night and I really, really need a cup of coffee. Just pretend I’m not me. I’m Mimi, a new customer.
Lorelai goes into Luke’s after an awful time with her parents, and asks for a coffee, suggesting he think of her as Mimi, a new customer. The assumed name Mimi sounds like me-me, and feels like a slight callback to Taylor calling Lorelai “Mini-Me”.