All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
First of all, we ask Allah to grant you success, increase you in guidance, and grant you lawful pure earnings.
Bank interests are Islamically forbidden because they are Riba (interest), which is one of the most destructive major sins.
As regards your family and you benefiting from the money of your relatives who take Riba, then you are not sinful for that – especially given that you are in need.
The prohibition here applies to the way in which the money is earned, and not to the money itself; hence, its prohibition applies to the person who earns it, not to his dealings with others with that money.
Ibn ‘Uthaymeen said, “The money that is unlawful because of the manner it was earned is like the money taken from Riba. It is forbidden for the person who earned it but not for the person who earns it from the former through lawful means. The evidence for this is that the Prophet dealt with the Jews, and it was well-known that they consumed unlawfully usurped wealth and dealt with Riba. This proves that it is not forbidden for a person other than the one who earned.” [Abridged from his Tafseer (exegesis) of Soorah Al-Baqarah]
However, you must advise them and remind them that those bank interests (Riba) are forbidden.
As regards the ruling on your mother depositing her money in the bank and taking the bank interests for herself, then this is totally unacceptable. If a person takes bank interests for himself, then he is consuming Riba, and consuming Riba (through lending) is not permissible at all, even at the time of necessity; contrary to borrowing money and paying interest (Riba) to the lender; this is a sound reason for its permissibility at the time of necessity.
The second conference for the Islamic Research Academy held in Cairo in the month of Muharram 1385 AH, corresponding to May 1965, which included delegates and representatives from thirty-five Islamic countries at the time of the prominent scholar Hassan Ma’moun, the grand Shaykh of Al-Azhar, unanimously resolved, “Lending money with Riba is forbidden, and no need nor necessity makes it permissible; borrowing money with Riba is also forbidden, and its sin is not removed unless there is a necessity for it.”
It should be noted that what is meant here are the interests, but the profits from the investment in Islamic banks are permissible.
For more benefit on the difference between forbidden interest and lawful profits, please refer to fatwas 85033 and 86266.
Allah knows best.