Lycra is the fiber that makes a suit stretchy. It's neither polyester or nylon, it's spandex. Lycra gets blended with other fibers (nylon, polyesters, cotton, wool, rayon etc) to create stretchy fabrics.
I think what she is trying to say is that you might have an allergy to either nylon, polyester or spandex. Elimination may be the only way to figure it out, and it will very difficult to find something that doesn't have spandex in it.