When is a son considered a mahram to his mother, and is it permissible for a twelve-year-old son to perform Hajj or Umrah with his mother and thus relieve her of the need for a mahram?

The Mahram with whom a woman travels must be an adult and of sound mind, because the purpose of the Mahram is to protect the woman, and this purpose is not achieved except by an adult of sound mind. Therefore, it is not permissible for a woman to travel for Hajj or otherwise, except with an adult Mahram of sound mind, or an adult husband of sound mind.