What is incumbent upon my mother, who swore that there were no clothes for her relative to wash, while her relative found the clothes and washed them, and is expiation due upon her?

If the mother intended the oath and made it binding, then she owes an expiation for the oath: feeding ten poor persons, or clothing them, or emancipating a slave. If she cannot find any of these, then fasting for three days. However, if the oath was uttered by her without intention, or if her intention was to honor the hostess, then there is no expiation due from her, and this is considered a 'laghw' (unintentional) oath.