Should the children of the deceased give their cousin a sum of money that their father had promised her to help with her marriage, knowing that it was neither a vow nor a formal promise, and that the relationship between them ceased after the father's death?

The children are not obligated to fulfill their father's promise to his niece, but it is desirable for them to do so, as it is a form of honoring the father after his death. If the promise was for something obligatory, then it must be fulfilled. If it was for something recommended, then it is recommended. Cutting off family ties should not lead the children to treat them in kind.