Is it permissible to give zakat to a mother in need, knowing that the son is unemployed and saving money for marriage and cannot provide for her?

It is not permissible to give Zakat (definition) to poor parents because their maintenance is obligatory upon the son, unless the son is unable to provide for them. In this case, he may give them Zakat because the obligation of maintenance is lifted from him due to his inability.

Therefore, if your mother is in need of money for her essential needs due to her and her husband's inability to provide for them, and you are unable to support her, then you may give her Zakat. Similarly, if she needs funds for matters beyond basic maintenance that are not obligatory upon her son or her husband, she may be given Zakat in that case.