This sounds as if the third eyelid has extruded from its normal anatomical position. While not a problem by itself, it may hint at underlying trouble such as upper respiratory infections, pulmonary disease, feline leukemia, and the list goes on.
This symptom is not unique to any one problem. Be aware that minor illness may also cause this third eyelid extrusion, and even in cats that have no discernable illness whatsoever, so don't panic.
A visit to your veterinarian, along with the kitten, will address the extent of the problem in your ten year old feline, and if in fact you have a problem at all.
Dr. Van Lienden
Dr. Raymond Van Lienden DVM
The Animal Clinic of Clifton
12702 Chapel Road, Clifton
Virginia, U.S.A. 20124