Is the mother obliged to offer expiation for her oath, by which she swore that her son's children would not remain in her house, after she accepted their stay out of fear of problems, and is she required to inform her son of the oath?

The breaking of an oath depends on the intention of the swearer and what is known as "the context of the oath." If the mother's intention behind her oath was to have a disagreement with her son, then the disappearance of that disagreement and reconciliation negates the ruling of the oath, and nothing is incumbent upon her. However, if this was not her intention, then she must offer expiation for the oath, which is feeding ten poor persons, or clothing them, or freeing a slave. If she cannot find any of these, then she must fast for three days.