How can one deal with a mother's incitement of her daughter (13 years old) to befriend a boy from the neighborhood, especially since the sister has just started to commit to prayer and righteousness?

Thank you for your concern and diligence regarding your sister, and may Allah reward you with good. We advise you to offer sincere counsel to your sister and mother in a gentle manner, emphasizing your concern for the family's reputation. Seek help from Allah Almighty and from people of virtue. If it is possible for you to move in with them, do so, and be a good role model.

If counsel does not work, and you believe that boycotting her will deter her, then boycott her. If you have guardianship over her, you can threaten and warn her by depriving her of studies or leaving the house except for necessity, or depriving her of these if it serves as a remedy for the situation. Beware of addressing a wrong with a greater wrong, or of being undutiful to your mother.