Can my mother give Zakat to her son, while my father is alive but very poor.
Zakat cannot be given to parents, grandfather, etc., or to one's children and grandchildren. Good to know, a husband and wife also cannot give Zakat to each other.
You may see that most of the scholars avoid giving any evidence regarding this. And as far I know, there is no direct verse in the quran and hadith. Then from where this prohibition came?
One thing to be understood that Zakat should be paid to the poor only. In Islam, if you are able then it is your duty to pay your direct ancestors and direct decedents, specially when they are in need.
So, without fulfilling their rights, how would you become eligible for paying Zakat?
You should pay them first then the zakat should be calculated on the rest of your wealth.
Well first of all usually it's not allowed to give zakat to children as the parents should care for their children so one of their duties to support them and the same is for children if they need to support their old or poor parents.
On the other Hand those whom should receive zakat are well known according to (9:60) and their you can read:
Zakah expenditures are only for the poor and for the needy ...
so poor people and those in need are in the highest category and should be the first to take into account when giving zakat. On the other hand it's a husband's duty to support his wife as good as he can according to Quran (65:7) and Sunnah
... Their rights upon you are that you should provide them with food and clothing in a fitting manner...
... and according to the following Fatwa (Arabic) a wife is neither asked nor should be forced to support the family of her husband even if she was wealthy. But she could do it to gain reward أجر by Allah.
So scholars say that in case of poverty and if children or parents couldn't afford their duties towards the other it is permissible to give either of them them zakat according to the mawso'a al fiqhya (Arabic) and here's an other Fatwa (Arabic).
What i really can't judge is whether the son or the father is in more need so according that zakat would be permissible to either of them!
And Allah knows best!