What is the ruling regarding a woman who swore an oath by Allah to her brother that he would not go on a certain trip, intending to scare him, then her father insisted that he go; what should she do?

It appears that what transpired was not a slip of the tongue, and the oath is binding. Since the oath has been broken, you must offer an expiation for breaking an oath, which is feeding ten poor people from the average of what you feed your families, or clothing them, or freeing a slave. And whoever cannot find that, then fasting for three days.