Seeking barakah from this hair is not permissible and it is a means that may lead to shirk, because it is impossible to prove that it is actually a hair of the Prophet (blessings and peace of Allah be upon him). By the same token, it is not permissible to build a mosque for it and keep it in that mosque, even if we assume for argument’s sake that it is actually a hair of the Prophet (blessings and peace of Allah be upon him) and is proven by an unbroken, sound chain of narration.