When June's father and grandmother delivered her to Marie, fresh from wandering the woods alone and eating tree sap to survive, she was wearing rosary beads as a necklace—and she refused to take them off. According to Marie:
Trying to explain to her that they were holy beads, not mere regular jewelry, did no good. She just backed away and clutched them in her fist. She wore them constant, even though the others teased her and jerked them lightly from behind when I was not looking. (5.1.23)
These beads were basically a security blanket for June, it seems, but she did ultimately give up the beads when she went to live with her Uncle Eli. She left them hidden in Marie's jar of odds and ends—and Marie, who had been having a rough time with Nector's hard partying ways, ended up using them for comfort herself.
You see, even though Marie isn't one for prayer, she liked touching the beads for reasons that she doesn't entirely articulate:
It has become a secret. I never look at them, just let my fingers roam to them when no one is in the house. It's a rare time when I do this. I touch them, and every time I do I think of small stones. At the bottom of the lake, rolled aimless by the waves, I think of them polished. To many people it would be a kindness. But I see no kindness in how the waves are grinding them smaller and smaller until they finally disappear. (5.1.94)
So, while each woman probably had her own individual idea of what the beads meant, the important takeaway here is that they represent a form of solace and comfort in the face of overwhelming suckiness.
Also, the beads bond June and Marie. Even though Marie was never really able to break through June's walls for too long, they are linked by blood—and the fact that they seemed to have more than their fair share of cruddy things to deal with. June was never particularly loving with Marie, but it seems like leaving the beads for Marie was a warm gesture—maybe she was passing off something that had given her comfort to someone who needed it more?