What should a son do for his deceased mother, how is the "Salat al-Hajat" (Prayer of Need) performed, and is it permissible to give charity on her behalf by purchasing Qur'ans for mosques?

The reward for patiently enduring the loss of a mother is Paradise. Prayer on behalf of the deceased is not valid. However, the reward for righteous deeds can be gifted to the deceased, such as charity, Hajj, Umrah, supplication (du'a), and seeking forgiveness (istighfar). As for seeing the mother in a dream, if it is considered a sign for performing Umrah on her behalf, there is no objection to that.