What is the expiation for one who falsely swore upon the Qur'an to clear his sister of a theft accusation?

What we understand from the question is that you swore a false oath on the Qur'an to clear your sister of a theft accusation. Swearing falsely is a grave sin, and you are obligated to repent to Allah Almighty. Al-Shafi'i and some scholars are of the opinion that expiation (kaffarah) is obligatory, and combining repentance with expiation is safer. Since you did this to save your sister from injustice, there is no blame on you for this oath, but you should offer expiation as a precaution.