Should she give the fidyah to her children and grandchildren or others as an iftar meal?

Firstly: 

If a person is unable to fast because of old age or sickness from which there is no hope of recovery, then he is allowed not to fast, but he should feed one poor person for each day, because Allaah says (interpretation of the meaning): 

“And as for those who can fast with difficulty, (e.g. an old man), they have (a choice either to fast or) to feed a Miskeen (poor person) (for every day)”

[al-Baqarah 2:184]

‘Abd-Allaah ibn ‘Abbaas (may Allaah be pleased with him) said: This has not been abrogated, it refers to an old man or an old woman who cannot fast; they should feed one poor person for each day. Narrated by al-Bukhaari, 4505. 

This also applies if a person is sick and there is no hope of recovery, as in the case of an old man who cannot fast. Al-Mughni, 4/396 

Hence it may be understood that this fidyah (ransom) is to be given only to the poor, not to everyone. 

If the children, grandchildren and students mentioned in the question are rich and are not poor, it is not permissible to give this fidyah to them. 

Secondly: 

As for giving this fidyah to her children and grandchildren, the scholars consider this fidyah to be like zakaah; it is not permissible for a person to give it to those on whom he is obliged to spend. 

Among those on whom one is obliged to spend are ascendants and descendants. 

Ascendants are the father, the mother, the grandfathers and the grandmothers. 

Descendants are sons and daughters and their children. 

Ibn Qudaamah said in al-Mughni (11/374): 

It is obligatory to spend on grandfathers and grandmothers, no matter how far the line of ascent reaches, and on one’s children’s children, no matter how far the line of descent reaches. This is also the view of al-Shaafa’i and al-Thawri, and of ashaab al-ra’i. End quote. 

Based on this, it is not permissible to give the fidyah mentioned here to children and grandchildren, because your mother is obliged to spend on them. 

Al-Shaafa’i said in al-Umm (7/68): 

It is not permissible to give food as expiation for a vow to anyone except a free Muslim who is in need; if the food is given to a dhimmi (non-Muslim living under Muslim rule) who is in need, or to a Muslim who is not in need, that does not count and the ruling is the same as one who did not do that; in that case he has to repeat it. The same applies if he gives food to one on whom he is obliged to spend. End quote 

It says in Asna al-Mataalib (3/369): 

With regard to the poor and needy, they are regarded as being among those who are entitled to zakaah, and it does not count if it is given to a kaafir… or to one on whom one is obliged to spend… because the kafaarah (expiation) is a duty that is owed to Allaah, so the same conditions apply as with zakaah. End quote. 

But if your mother cannot afford to spend on them, because she is not rich, then she is not obliged to spend on them, because Allaah says (interpretation of the meaning): 

“Allaah burdens not a person beyond his scope”

[al-Baqarah 2:286]

In this case, it is permissible for her to give this fidyah to them. 

It is proven in al-Saheehayn that the Prophet (peace and blessings of Allaah be upon him) said to a man who had intercourse with his wife during the day in Ramadaan, when the Prophet (peace and blessings of Allaah be upon him) gave him some dates to give as an expiation and the man told the Prophet (peace and blessings of Allaah be upon him) that he was the poorest household in Madeenah – the Prophet (peace and blessings of Allaah be upon him) said to him, “Give them to your family.” 

Al-Haafiz said in al-Fath: 

Ibn Daqeeq al-‘Eid said: There are different views concerning this story. It was said that it indicates that the kafaarah (expiation) is waived in cases of hardship, because the expiation cannot be given to one’s own self or one’s dependents. 

The majority said that the expiation is not waived in cases of hardship, and that what he was given permission to do was not by way of expiation, rather it was charity that was given by the Prophet (peace and blessings of Allaah be upon him) to this man and his family. 

And it was said that because he could not afford to spend on his family, he was permitted to give this expiation to them. This is the apparent meaning of the hadeeth. Shaykh Taqi al-Deen (i.e., Shaykh al-Islam Ibn Taymiyah) said: A stronger view is that it was given not as an expiation, but as a charity to him and his family, because it is apparent that they were in need. End quote. 

From the above it may be understood that it is not permissible to give the expiation to those on whom one is obliged to spend, but if a person is poor and cannot spend on them, then some of the scholars are of the view that it is permissible to give the expiation to them. 

In the answer to question no. 20278 we have quoted the fatwa of Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) which says that it is permissible to give zakaah to relatives on whom one cannot spend because of poverty and lack of wealth. 

It says there: If he gives the zakaah to his relatives who are entitled to zakaah, that is better than giving it to those who are not relatives, because charity given to a relative is both charity and upholding the ties of kinship. 

But if these relatives are among those on whom one is obliged to spend, and you give them zakaah so that you will not have to spend on them from your wealth, this is not permissible. 

But if you cannot afford to spend on them, then there is no sin in giving them your zakaah. End quote. 

In conclusion: If your mother is rich and can afford to spend on them, then it is not permissible for her to give them her fidyah or expiation. If she cannot afford to spend on them, it is permissible for her to give them her fidyah or expiation. 

Thirdly: 

If she gives it by way of offering iftaar to one who is fasting, there is nothing wrong with that, because of the general meaning of the verse: “to feed a Miskeen (poor person)”. There is the hope that this will bring a greater reward, because it is giving iftaar to one who is fasting. But that is subject to the condition that the fasting person be poor, as stated above.