Praise be to Allah.
There is nothing wrong with your father asking his brother to give him his full rightful share of the estate, because the estate of the deceased is a trust for which the one who takes charge of it is responsible, and he must give it to the heirs listed in shar‘i texts, each according to the shares that have been ordained by Allah, may He be glorified and exalted. If a person seeks his rights, there is no blame on him, either in this world or in the hereafter. Allah, may He be glorified and exalted, says (interpretation of the meaning):
“Allah does not like that evil should be uttered in public except by him who has been wronged. And Allah is Ever All-Hearer, All-Knower
Whether you (mankind) disclose (by good words of thanks) a good deed (done to you in the form of a favour by someone), or conceal it, or pardon an evil, verily, Allah is Ever Oft-Pardoning, All-Powerful”
The one who has been wronged should have the right to demand his rights from the one who wronged him, by referring the matter to the qaadi (judge), and to complain to people about the wrongdoing done to him, according to the text of the Holy Qur’an.
However we should point out a number of important matters to which attention should be paid before referring the matter to the sharee‘ah courts; these include the following:
It is essential to certain that wrongdoing has in fact occurred before demanding one’s rights. This requires clear proof, and is not to be based on doubts, conjecture or malicious gossip that is spread by some people, especially if the estate was divided by the sharee‘ah courts.
It is essential for there to be a calm discussion between your father and his brother; he should complain to him directly so that each of them can hear the other’s point of view. At the same time, good, knowledgeable and righteous people should also intervene in order to explain matters. You should not give up on finding a solution to the problem before referring the matter to the courts. The courts should be the last resort, after patiently looking for a solution.
There is no sin on your father if his mother gets upset or angry because of his going to the court. Rather she has no right to get upset if the wrong that was done to him is clearly established and he has undeniable proof to that effect. In that case it is essential to convince the mother about this proof and to calmly discuss the matter with her, so that she will accept it and understand the causes that led to that. Then, if she continues to be upset and angry, there is no sin on your father, because his mother has no right to ask her son to give up his rights or to keep quiet about the wrong that has been done to him.
Ibn Taymiyah (may Allah have mercy on him) said:
The individual is required to obey his parents in any matter that does not involve sin, even if they are evildoers. This is the apparent meaning of the statement of Ahmad. This has to do with matters that are of benefit to them and do not cause him any harm; if it is difficult for him but will not harm him, it is obligatory, otherwise it is not.
End quote from al-Ikhtiyaaraat al-Fiqhiyyah by al-Ba‘li (114)
If keeping quiet about his rights will clearly be harmful, then obedience to his mother is not obligatory in this case, and he does not have to give in to his brother and let him off in order to please his mother, even though doing so (pleasing one’s parents) is best in all cases, as Allah, may He be glorified and exalted, says:
“Let them pardon and forgive. Do you not love that Allah should forgive you? And Allah is Oft-Forgiving, Most Merciful”
Whoever seeks high aims, we advise him to disregard what has happened and to pay attention to his mother's likely distress if she sees her sons disputing with one another. At the same time, the one who refers the matter to the courts is not to be blamed.
And Allah knows best.