Ms. Rich was cast because of her big expressive eyes- Curtis wanted a little girl that would, in some fashion, evoke Grayson's Carlotta Drake. Think of the great large close-ups of Grayson's face, highlighting her eyes.
There was an ad for a charity in a national magazine, I believe LIFE or something similar in the months prior to NODS shooting, featuring young Ms. Rich as sad child looking forlorn with expressive eyes. According to her, that's what got her cast in the film. She also had done a little bit of acting- commercials & the like, I believe and a PBS operetta "MY HEART IS IN THE HIGHLANDS," which both Museum's of TV & Radio in NY and LA have copies for viewing.