Is it permissible for the questioner's mother to give her Zakat al-mal to her nephew who is studying abroad at his own expense, knowing that his father is financially well-off but does not help him?

The recipients of zakat (definition) are specified in the verse: "Indeed, zakat expenditures are only for the poor and for the needy and for those employed to collect [zakat] and for bringing hearts together [for Islam] and for freeing captives [or slaves] and for those in debt and for the cause of Allah and for the [stranded] traveler - an obligation [imposed] by Allah. And Allah is Knowing and Wise." Your mother may give her zakat to your maternal cousin if he falls into one of these eight categories. A student of worldly knowledge is not given zakat by virtue of seeking knowledge, nor is it permissible to give zakat to one who is able to earn a living, due to the Prophet's (peace be upon him) saying: "Charity is not permissible for the rich nor for one who is strong and able to earn a living."