What should be done with the money of an elderly shaykh who died and left no heir except for a maternal half-sister, with his other half-siblings having passed away before him?

Your grandmother is the deceased's maternal half-sister, and she inherits all that he left, by both fard (prescribed share) and radd (return of surplus), due to the absence of any other heir. Thus, she receives one-sixth as a fard, and the remainder is returned to her. You are not permitted to take any of the money without her permission. If she donates any of it for building a mosque or for other purposes, that is good. The matter of inheritances is serious and complex, and the case must be referred to the Sharia (definition) courts to investigate the heirs, wills, debts, and other rights before the estate is distributed.