Reading the Holy Qur’aan at the graveside is bid’ah. It was not narrated from the Prophet (peace and blessings of Allaah be upon him) or from his companions. As it is not narrated from the Prophet (peace and blessings of Allaah be upon him) or from his companions, then we should not introduce it by ourselves, because the Prophet (peace and blessings of Allaah be upon him) said, according to a saheeh report: “Every introduced matter is an innovation, and every innovation is a going astray, and every going astray will be in the Fire.” What the Muslims must do is to follow the example of the earlier generations, the Sahaabah and those who followed them in truth, so that they will be following goodness and true guidance, because it is proven that the Prophet (peace and blessings of Allaah be upon him) said: “The best of speech is the Book of Allaah and the best of guidance is the guidance of Muhammad (peace and blessings of Allaah be upon him).”
As for saying du’aa’ for the deceased at the graveside, there is nothing wrong with that. A person may stand at the graveside and say whatever du’aa’s he can, such as saying, “O Allaah, forgive him; O Allaah, have mercy on him; O Allaah, admit him to Paradise; O Allaah, make his grave spacious” and so on.
With regard to a person saying du’aa’ for himself at the graveside, if he does it deliberately then it is also an innovation (bid’ah), because no place is specified as a place for du’aa’ unless there is a text concerning it. If there is no text concerning it and no Sunnah has been narrated concerning it, then that – i.e., singling out a place for du’aa’, no matter what the place is – is a bid’ah. End quote.
Majmoo’ Fataawa Ibn ‘Uthaymeen (17/228)
For more information please see the answer to question no. 36513.
And Allaah knows best.