They believed this because they read her tea leaves. Their prediction turns out to be true since Coraline is the Other Mother's newest target. Late in the movie, she used it to find the eyes of the three ghost children. She throws it at a rat that had the third eye, but the rat dodges it and became lost. Later on The Other Mother revealed she had it then burned it trying to prevent Coraline from finding her parents.
In the book, the Stone was sort of a protective amulet against evil magic. While Coraline has it, it keeps her from falling completely under the Other Mother's power, and at times the Other Mother seems to be repulsed by it. In both the book and the film, the Stone allows Coraline to find the lost souls of the Ghost Children when she looks through the Stone's hole. In the book she keeps the Stone with her long enough to return it to Miss Spink and Forcible when she no longer needs it.
The stone Coraline use may have been a adder stone, an magical item allow the user to see and find magic items and fairy creatures.