LADAWN: Welcome to the Cheshire Cat.
LORELAI: … Okay, she’s named the place after an Alice in Wonderland character. This is my worst nightmare.
As Lorelai says, the Cheshire Cat is a character in Lewis Carroll’s Alice’s Adventures in Wonderland, previously discussed and frequently referenced. The cat likes to engage in amusing philosophical conversations, and is known for its distinctive grin; even when it disappears, as it does a few times, the grin will be the last thing to vanish.
The character is a personification of the English saying, “to grin like a Cheshire cat”, meaning that the person has a very wide smile. The saying dates to the 18th century, but its origins are obscure. Cheshire was known for its dairy industry, with milk and cream making cats happy, so that’s one possible explanation.
Presumably LaDawn chose the name of her B&B after her large cat, Sammy. In real life there are numerous B&Bs in the Portsmouth area to cater to the tourist trade. Scenes at the Cheshire Cat were filmed in the same set used for the Black White Read Bookstore.