Is it a sin to say, “I swear on my mother’s grave”?
No, because invoking the sacred to guarantee one’s word is a time-honored custom. A common example of this is to swear one’s oath upon a Bible in a court of law.
However, one should only invoke the sacred in oaths when there is just cause to do so, such as for solemn occasions in civic life (e.g., inauguration, testifying in court, becoming a citizen). On lesser occasions, it is best to follow Jesus’ advice for giving one’s word:
But I say to you, “Do not swear at all, either by heaven, for it is the throne of God, or by the earth, for it is his footstool, or by Jerusalem, for it is the city of the great King. And do not swear by your head, for you cannot make one hair white or black. Let what you say be simply ‘Yes’ or ‘No’; anything more than this comes from evil”