Praise be to Allah.
This question covers a number of topics:
1 – Those who are entitled to inheritance in this case are:
The father and mother, and the husband and wife, and the sons, if there are sons, are not prevented from inheriting. But each of them should take the share which is decreed by Allaah. But in the case of daughters, if there are also sons, they inherit by right of descent [bi’l-ta’seeb] (The word ‘by right of descent’ can be used with the sons and daughters but notwith the brothers, sisters, uncles and cousins. Therefore, I would think thatthe word ‘by agnation’ is more suitable because it means the blood relationshipfrom the side of the father ) on the basis that each male gets the equivalent of the share of two females.
In the case you mention, those who are entitled to inheritance are:
The husband, who is entitled to one-quarter because there is a son (a descendent who is entitled to inherit)
The son, who is entitled to the remainder of the inheritance, because he inherits by right of descent [ bil-ta’seeb ]
Brothers and sisters are prevented from inheriting by the presence of the son, so they have no right to the inheritance.
2 – Ruling on making a will in favour of an heir:
It is not permissible to make a will in favour of any of the heirs (i.e., those who are entitled to inherit according to sharee’ah). It was narrated in a saheeh report that the Prophet (peace and blessings of Allaah be upon him) said: “Allaah has given to each one who is entitled his rights, and there is no will in favour of one who is an heir…” (Narrated by al-Tirmidhi, 2047; classed as saheeh by al-Albaani in Saheeh Abi Dawood, no. 1722.) Shaykh Taqiy al-Deen said, The ummah is unanimously agreed upon this.
Unless that will is approved by the heirs (who are entitled to inherit according to sharee’ah), in which case it becomes valid; it is subject to their consent.
3 – When the will is to be considered.
The will is to be considered immediately following death.
Al-Muwaffaq said: we do not know of any dispute among the scholars concerning the fact that the will is to be considered immediately following death.
See al-Mulakhkhas al-Fiqhi by al-Fawzaan, part 2, p. 174.
Ibn Hajar said: The (scholars) are agreed that the one in whose favour the will is made is to be considered as an heir (i.e., one who is entitled to inherit according to sharee’ah) on the day of death. Even if one made a will in favour of a brother who is an heir (i.e., one who is entitled to inherit according to sharee’ah), as he does not have a son to prevent his brother from inheriting, but just before he dies a son is born to him, and that son prevents his brother from inheriting, then this will in favour of the brother is valid. If he made a will in favour of his brother at the time when he had a son, and the son died before his father, then the brother becomes entitled to inherit and thus the will becomes invalid.
On this basis it is permissible for your mother to make a will in favour of her two grandchildren (your two children) so long as they do not have any rights of inheritance. She may also include her brothers and sisters in her will, on the condition that she does not give more than one-third to them, because the Prophet (peace and blessings of Allaah be upon him) said: “One-third, and one-third is a lot.” (Narrated by al-Bukhaari, al-Wasaayaa, 2538)
With regard to giving something whilst one is still alive, this is a gift. See the answer to Question No. 10447