If a son knows that his mother purchased tools with ill-gotten money, is he obligated to sell them and donate their price in charity?

There is a difference between money that is entirely unlawful and mixed money. If your brother's money is entirely unlawful, scholars have differed as to whether its prohibition is limited to its owner or extends to others. According to the view that the prohibition extends, you are permitted to use what you are compelled to out of necessity from tools due to your poverty. As for what is not necessary, you should dispose of it or its value in charitable causes and public welfare.