I know that lottery money is haraam. My question is: is it permissible to use it to pay off my debts to the bank and my siblings’ debts to the bank, so long as I am careful not to use it for anything else?.
The lottery is a kind of gambling, and money acquired by means of it is haraam wealth which must be gotten rid of by spending it on the poor and needy and other charitable causes. It is not permissible for the individual to benefit from it himself, and he also has to repent to Allah, may He be exalted, from what he has done.
Allah, may He be exalted, says (interpretation of the meaning):
“O you who believe! Intoxicants (all kinds of alcoholic drinks), and gambling, and Al‑Ansaab (stone altars for sacrifices to idols etc), and Al‑Azlaam (arrows for seeking luck or decision) are an abomination of Shaytaan’s (Satan’s) handiwork. So avoid (strictly all) that (abomination) in order that you may be successful.
91. Shaytaan (Satan) wants only to excite enmity and hatred between you with intoxicants (alcoholic drinks) and gambling, and hinder you from the remembrance of Allaah and from As‑Salaah (the prayer). So, will you not then abstain?”
See also the answer to question no. 6476.
Based on that, it is permissible to give it to your siblings so that they may benefit from it by paying off their debts or buying what they need if they are poor.
If you do not have enough money to pay off your debt, then it seems that it is permissible to you to take some of this money to pay them off. Ibn al-Qayyim (may Allah have mercy on him) spoke in detail about the issue of getting rid of haraam wealth, and he stated that the way to get rid of this wealth and complete one's repentance can only be by “giving it in charity, but if he is in need of it, he may take as much as he needs, and give the rest in charity. See Zaad al-Ma‘aad, 5/778
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: If this prostitute and this wine seller repent, and they are poor, it is permissible for some of this money to be given to them, as much as they need. Af he is able to engage trade or a profession such as weaving or spinning, he should be given enough to form capital for him. End quote from Majmoo‘ al-Fataawa (29/308)
And Allah knows best.