What is the ruling on specializing Surat Ar-Ra'd for treating headaches and writing it on a cap and wearing it, and what should be done with a mother who compels her daughter to wear it if that is an innovation?

There is no specific evidence regarding particular virtues of Surah Ar-Ra'd. What is mentioned in this regard falls under the science of the properties of the Quran, about which scholars differ between considering it an innovation or permitting it based on experience.

It is permissible to recite the Quran for worship, for ruqyah (exorcism/healing), and to ask Allah for all needs through it, as confirmed by the Prophet, peace and blessings be upon him: "Recite the Quran and ask Allah by it."

As for writing the Quran and hanging it as an amulet, most scholars consider it impermissible due to the general prohibition of amulets, to block the means to hanging things that are not from the Quran, and because they are often not safe from disrespect.

If the questioner believes that hanging an amulet is forbidden, she should not obey her mother in that regard. She should discuss it gently with her mother to explain the jurisprudential ruling, while continuing to honor and be good to her. She should apologize for not obeying in a matter that is forbidden, based on the Prophet's saying, peace and blessings be upon him: "Obedience is only in what is good and lawful." However, if the matter is unclear to her, then she should obey her mother.