Rebecca Nurse, by her own admission, is a woman who is very experienced with looking after children. She has a number of her own children and grandchildren, and has been involved in taking care of them all. It is thus perhaps no surprise that it is she who is able to calm Betty and make her go to sleep. Note what she herself says about Betty and what is really troubling her:
A child's spirit is like a child, you can never catch it by running after it; you must stand still, and, for love, it will soon itself come back.
It is interesting to reflect that this advice has not been carried out by the other characters, who have all not been truly worried about Betty but have only been worried about how her mysterious illness will reflect on them and might get them into trouble. Rebecca Nurse is the only person who truly loves Betty and is not worried about their own position. Rebecca treats Betty's illness not as being the result of witchcraft but being something completely natural.
She is the only one in the room who realises that witch craft isn't the cause of Betty's condition. She says "There's a hard sickness here, Giles" Which means that Betty is ill no under a spell.