Is it permissible for a person who told his mother, "I will not speak to so-and-so," or "I will not do such-and-such" to please her, to do so after her death, and is his statement "I will not do" considered a promise or a covenant, and is it permissible to break it?

The son is not obligated to abide by his parents' commands after their death, and disobeying them is not considered an act of disobedience. Thus, it is even more fitting that he is not obligated to fulfill what he promised his mother after her death. Furthermore, obedience to parents is not obligatory in matters involving disobedience, such as abandonment.