Should I continue studying Islamic law at the mosque's course, despite my mother's objection and insistence on me staying at home, or should I stop to please her?

There is no harm for a daughter to go out to seek religious knowledge. If acquiring this knowledge is obligatory and there is no way to obtain it except by going out, then it is not permissible for the mother to prevent her daughter from going out. The daughter must acquire the knowledge, even if it leads to her mother's displeasure.

However, if the knowledge is supererogatory (recommended) or can be acquired at home, then she should not go out except with her mother's consent. If the mother insists on refusing, the daughter should not go out but should acquire the knowledge through other available means, such as listening to lectures and reading fatwas and publications online.