Their father has died, and they live in the West where the courts operating under man-made laws give the mother half of the marital home; what is the ruling?
Your mother’s share of your father’s estate, including the house and whatever else he left behind, according to sharee‘ah, is one eighth only, because Allah, may He be exalted, says (interpretation of the meaning):
“their (your wives) share is a fourth if you leave no child; but if you leave a child, they get an eighth of that which you leave after payment of legacies that you may have bequeathed or debts”
This is a matter on which there is scholarly consensus, and it is something well known among the ummah.
It is not permissible for the Muslim, male or female, to demand another share, other than the share allocated by Allah, may He be glorified and exalted, seeking to increase his share or to decrease the share of someone else. Allah has given a warning of a painful punishment for the one who transgresses His limits with regard to shares of inheritance or other shar‘i rulings. Allah, may He be exalted, says after mentioning the shares of inheritance (interpretation of the meaning):
“These are the limits (set by) Allah (or ordainments as regards laws of inheritance), and whosoever obeys Allah and His Messenger (Muhammad blessings and peace of Allah be upon him) will be admitted to Gardens under which rivers flow (in Paradise), to abide therein, and that will be the great success.
And whosoever disobeys Allah and His Messenger (Muhammad blessings and peace of Allah be upon him), and transgresses His limits, He will cast him into the Fire, to abide therein; and he shall have a disgraceful torment”
What you must do is explain that shar‘i principle to your mother, and tell her that it is haraam to transgress the limits set by Allah, and that this is an important principle of faith: the individual should be content with the ruling of Allah and what He allocates to His slaves, and not refer for judgement concerning any matter, small or great, to any man-made laws, traditions and customs. This is one of the important requirements of faith, that one should submit completely to the rulings of Allah and of His Messenger, whether or not that is in accordance with one’s whims and desires.
Allah, may He be exalted, says (interpretation of the meaning):
“But no, by your Lord, they can have no Faith, until they make you (O Muhammad blessings and peace of Allah be upon him) judge in all disputes between them, and find in themselves no resistance against your decisions, and accept (them) with full submission”
“They (hypocrites) say: "We have believed in Allah and in the Messenger (Muhammad blessings and peace of Allah be upon him), and we obey," then a party of them turn away thereafter, such are not believers.
And when they are called to Allah (i.e. His Words, the Quran) and His Messenger (blessings and peace of Allah be upon him), to judge between them, lo! a party of them refuse (to come) and turn away.
But if the right is with them, they come to him willingly with submission.
Is there a disease in their hearts? Or do they doubt or fear lest Allah and His Messenger (blessings and peace of Allah be upon him) should wrong them in judgement. Nay, it is they themselves who are the Zalimoon (polytheists, hypocrites and wrong-doers, etc).
The only saying of the faithful believers, when they are called to Allah (His Words, the Quran) and His Messenger (blessings and peace of Allah be upon him), to judge between them, is that they say: "We hear and we obey." And such are the prosperous ones (who will live forever in Paradise).
And whosoever obeys Allah and His Messenger (blessings and peace of Allah be upon him), fears Allah, and keeps his duty (to Him), such are the successful ones”
It is not permissible for you to resort to the courts that are based on man-made, kaafir laws; rather what you must do is go to the nearest trustworthy Islamic centre that is run by people of knowledge who follow the Sunnah. Perhaps that will help you to convince your mother and explain to her what is right, and that it is haraam to resort to man-made laws or to refer for judgement to their courts.
If it so happens that any of the procedures of transferring ownership, or buying or selling, require confirmation of the court for any of that, then what you must understand is that referring to these courts is a superficial matter only, and whatever verdict these courts issue, it is not permissible for you to approve of it, accept it or adhere to it, unless it is in accordance with the laws of Allah.
If any of the heirs wants to obtain his share, then he has the right to do so. In that case the house must be divided according to the shares allocated in sharee‘ah, then each of you should take his share. If anyone wants to give up his share or any part of it to his mother, then he has the right to do that, but it is not obligatory for the others to do so.
Rather what you must do with regard to your mother is provide her with accommodation that is appropriate for someone of her standing. If what she owns of wealth, from her share of the inheritance or if she has other wealth, is sufficient to buy suitable accommodation for her, then she has the right to buy it with her own wealth, and her sons do not have to buy that with their wealth, unless they willingly choose to do so.
If her wealth is not sufficient for that, then what you must do is provide her with suitable accommodation, but that does not have to be by buying; rather you may either help her to buy or help her to rent that accommodation, if her wealth is not enough to do that.
However the matter is as you say: she is your mother and the matter should not be settled between you by means of fighting through the courts. Rather what you must do is strive to please her and make things easy for her in whatever ways you can, in a way that is not detrimental to your own rights and wealth. Be kind with her and be gentle with her when explaining what is correct, and provide her with what she needs.
What the Muslim should do is strive to be honest in his dealings. He should understand that in such countries he should try to present a good image of Islam, as people will regard him as being a representative of his religion. So it is not appropriate for him to give them an image of Islam and its people that is an image of cheating, deceit and breaking promises, even if your honesty results in some kind of financial penalty or loss, or paying some kind of taxes, especially if the benefit of those taxes comes to all the people, which includes you, and is spent on public services, unemployment assistance, maintenance of facilities from which you benefit, and so on.
And Allah knows best.