Is it permissible for a divorced mother to file a lawsuit against her ex-husband for not providing for their son, even if her father disapproves, given that this is the son's right?

If your children are in need of financial support, their father is obligated to spend on them in a reasonable manner. If he refuses, it is permissible to resort to legal action to secure the children's right to maintenance. If the father prevents this, he is not to be obeyed, unless there is no other way to obtain the right except through the judiciary, especially if harm results from it, while still being keen on honoring and showing kindness to the father.