While talking with Lorelai, Babette mourns the death of her elderly cat Cinnamon, remembering the time when she was a tiny kitten. As she and Morey are a childless couple, Babette speaks of Cinnamon’s passing as if she and Morey have lost a child, even mentioning an Oprah episode about bereaved parents whose marriages didn’t survive.
Babette’s feelings for Cinnamon serve as a parallel to Lorelai’s for Rory’s (the cat and the girl are probably even of similar ages). I believe Cinnamon was changed from a male cat to a female one to make this parallel more obvious.
Lorelai must also face “losing Rory” one day – not to death, but to adulthood. She knows that Rory is growing up, and that she has to learn to let her go, and to find a way to make a life for herself that isn’t based around caring for Rory.
This helps to inform her decision about dating Max: Rory is growing up, and Lorelai can’t put her own life on hold forever.