What is the ruling on supplicating with the words: "O Allah, I ask You by the prestige of so-and-so" or "by the right of so-and-so," and does this differ from saying: "O so-and-so, aid me" to the occupant of a grave?

It is not permissible to ask Allah Almighty by virtue of anyone's status or right, even if they are from the prophets or saints, because Allah owes no one any right. One must only ask through Allah's names and attributes, as He says: "And to Allah belong the best names, so invoke Him by them." As for invoking the dweller of a grave, such as saying, "O so-and-so, help me," this is clear polytheism; because it is an invocation to other than Allah. Thus, asking by virtue of one's status is a means to polytheism, and invoking a creature is polytheism in worship.