What does Islam say about reading Qur'an on the dead? Is it permissible for any one to read on any dead (Muslim) person or should it be a family member?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Presenting the reward of recitation of the Qur'an to a dead person is permissible whether the dead is a relative or not. This is the opinion of the majority of Muslim scholars.
But, it is better to leave it, instead it is recommended to multiply making Du'a for a deceased and giving charity on behalf of him since such deeds are proved by the sound Sunnah. On the other hand, reciting the Qur'an and presenting its reward was not the practice of the Prophet's (Sallallahu Alaihi wa Sallam) companions in spite of the death of many Muslims and their relatives. Know that all the given details relate only to the matter of recitation of the Qur'an and presenting its reward.
As for reciting the Qur'an over a grave or when the people gather for offering condolences, both practices are innovations. Visiting graves is permitted only to remind oneself of the life Hereafter and to send greetings on dead persons. There is no evidence proving that the recitation of the Qur'an over graves is likeable. Holding mourning-gatherings to offer condolence and reciting the Qur'an in such gatherings is also not lawful.
Allah knows best.