Is it permissible for my mother to sue her siblings to claim her rightful share of the inheritance after the death of my grandfather, knowing that she took a small portion of it during his lifetime, and would she bear any sin for doing so?

It is permissible to relinquish an inheritance or part of it. It is preferable to accept a reduction in one's right if it is out of generosity and nobility, whereas one who withholds a right is an unjust transgressor, especially concerning inheritance. Such a person is warned by the Almighty's saying: "Those are the limits [set by] Allah, and whoever obeys Allah and His Messenger will be admitted by Him to gardens beneath which rivers flow, abiding eternally therein; and that is the great attainment. And whoever disobeys Allah and His Messenger and transgresses His limits – He will admit him to Fire, abiding eternally therein, and he will have a humiliating punishment." The injustice inflicted upon women is even more reprehensible. The Prophet, peace and blessings be upon him, said: "O Allah, I declare inviolable the rights of the two weak ones: the orphan and the woman." One who does not take their full right is not sinful; rather, the sinner is the one who withheld their right. The amount given by the grandfather to your mother before his death is not considered part of the inheritance and must be returned to the estate or deducted from her share, as it is either an invalid gift due to lack of fairness among the children or a debt that must be repaid. It is not permissible to deduct the living expenses of the daughters from their share of the inheritance. It is permissible to refer the matter to the Sharia court, but reconciliation and mutual agreement are preferable.