Praise be to Allah
Prayer is one of the greatest pillars of Islam after the Shahaadatayn (twin declaration of faith). What the Muslim must do is always offer the prayers regularly and on time; this duty is not waived so long as he is of sound mind.
For more information, please see the answer to question no. 95220.
You did not say whether your mother had become unconscious and thus was not aware of anything, or not. However it appears from your question that she was of sound mind at the time of her illness, based on the fact that she intended to make up those prayers. In general, the one who does not pray in the case of illness must come under one of the following two scenarios:
1.He stopped praying when he was sick because he had lost his mind or lost consciousness. In this case he does not have to do anything, in sha Allah, and he is not obliged to make up the prayers when he recovers after that.
2.He stopped praying when he was sick but was still of sound mind; but he stopped praying because he thought that prayer was not obligatory for him in that situation. In this case, perhaps Allah will pardon him because of his ignorance, although what the individual is required to do is to learn what he must know of his religion.
In either case, the prayers are not to be made up on his behalf if he dies after that.
It says in Fataawa al-Lajnah ad-Daa’imah (25/257):
If your father was unconscious at the time of his illness and did not understand anything, then the obligation of prayer was waived in his case, and he is not accountable when he is in this state. The condition of being accountable for prayer is that the individual should be of sound mind (and not unconscious), and that was not the case here. But if he was not unconscious and was of sound mind, but he did not pray because he was unaware that it is obligatory for people in his situation to pray according to what they are able to do, then perhaps Allah will pardon him and excuse him because he was unaware of that, and because there was no one who could explain the shar‘i ruling to him before he died – may Allah have mercy on him and forgive him. In either case, it is not permissible for you to offer any of the prayers on your father’s behalf, because one cannot pray on behalf of anyone else. The basic ruling is that prayer cannot be delegated to another person. End quote.
With regard to making up the fasts, if she did not make them up with no excuse, then it is mustahabb for you to fast on her behalf, because of the report narrated by al-Bukhaari (1952) and Muslim (1935) from the hadeeth of ‘Aa’ishah (may Allah be pleased with her), according to which the Prophet (blessings and peace of Allah be upon him) said: “If anyone dies still owing fasts, let his heir fast on his behalf.”
For more information, please see the answers to questions no. 130283 and 130647
It is permissible for making up of the fasts on behalf of the deceased to be done by one heir for all of the days, or it may be done by a number of heirs.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
Question: If we follow the more correct opinion, which is that fasting includes that which is obligatory because of basic Islamic teachings and that which is obligatory because of a vow (nadhr), does that mean that this (making up the fasts of behalf of the deceased) is to be done by one of the heirs only, because fasting was obligatory upon one person only?
Answer: It does not mean that, because the words of the Prophet (blessings and peace of Allah be upon him), “let his heir fast on his behalf” include all the heirs. So if we assume that a man has fifteen sons, and each of them wants to fast two out of thirty days, that is acceptable. If there are thirty heirs and each of them fasts one day, that is acceptable, because they will have fasted thirty days. It makes no difference whether they fast on one day, or one fasts on one day, then the second one fasts on the next day, (and so on) until they have completed thirty days.
End quote from ash-Sharh al-Mumti‘ (6/452)
If a person performs Hajj or ‘Umrah on behalf of his parents, and he has performed Hajj on his own behalf before that, then the reward of that Hajj or ‘Umrah will reach his parents, and that action on the part of the son will be a kind of honouring and kindness towards his parents.
Muslim (1939) narrated from ‘Abdullah ibn Buraydah that his father (may Allah be pleased with him) said: Whilst I was sitting with the Messenger of Allah (blessings and peace of Allah be upon him), a woman came to him and said: I gave a slave woman in charity to my mother, who has now died. He said: “Your reward is due; you may take (the slave woman) back by virtue of your inheritance.” She said: O Messenger of Allah, she owes one month’s fast; can I fast on her behalf? He said: “Fast on her behalf.” She said: She never did Hajj; can I do Hajj on her behalf? He said: “Do Hajj on her behalf.”
It says in Fataawa al-Lajnah ad-Daa’imah (25/257): With regard to your doing Hajj and ‘umrah on behalf of your father, that is part of honouring him and showing kindness to him. And if you give charity on his behalf from time to time, and you offer supplication for him, pray for forgiveness for him, uphold ties of kinship with his relatives, and treat his friends kindly, then that is a way of honouring your father after his death, and you will have a great reward, in sha Allah, for whatever you spend for that purpose. End quote.
For more information, please see the answers to questions no. 763 and 104606
And Allah knows best.