Someone else said a couple pages ago that the original Run was based off The Running Man and there were spandex costumes in that.
Not quite, it's a good theory but you gotta' think a little deeper of the connection between Run and Spandex.
Being one who's worn a spandex zentai/morphsuit for a haunt before, the connection is actually really simple. One thing you don't want to have in that kind of suit is a tear or rip...also known as a "RUN".