That "thing" is called the caruncle and it doesn't really do anything. It is a vestigeal third eyelid/nictating membrane (present in some lower animal/reptiles). There is a fold of conjjuctiva there also called the plica semilunaris, it doesn't do anything either.
JCH IIIMD
can cold weather make the eye dry? I have been in a hot climate for 6 months and am now in UK where it is very cold.