Do the children of someone who died during his mother's lifetime have a share in their grandmother's inheritance, knowing that their father had renounced his right to the inheritance—in exchange for financial assistance—before his mother's death?

There is no validity to a son's relinquishment of his right to inherit from his mother before her death. If the son dies before his mother, his inheritance from her isقطع (severed). Grandchildren do not inherit from their grandmother if she has male children, because sons block the inheritance of grandchildren.