My cat has developed a runny eye and his third lid is covering his eye slightly.
My first reaction was that it sounded like conjunctivitis. I have a page on the eye problems that are most common with cats and will provide the link here. This page will have additional links to individual cat eye problems and diseases so that you can peruse them and decide which best fits the symptoms that you are noticing with your cat. There are several which involve the third eyelid showing.