A receivable debt remains part of the lender's property and is payable even if the lender dies. If the lender dies, all his wealth, including receivables, is the property of the heirs (according to their shares) and should be distributed among them.
Quran 4:33 And for all, We have made heirs to what is left by
parents and relatives. And to those whom your oaths have bound [to
you] - give them their share. Indeed Allah is ever, over all things, a
Sahih Bukhari: The Prophet (ﷺ) said, " If somebody dies (among the
Muslims) leaving some property, the property will go to his heirs; and
if he leaves a debt or dependents, we will take care of them."
If the person to whom the debt is owed has died, then what the debtor
must do, when he is able to, is to repay the money to his heirs, or
ask all of them to let him off, or at least ask each heir to let him
off his share of this money. The Prophet (blessings and peace of Allah
be upon him) said: “Whoever leaves behind wealth, it is for his
heirs.” Narrated by al-Bukhaari (2297).
This money should be given to all of the heirs, or their deputy, if
they have appointed one, or their guardian, if they are not yet
The seriousness of settling a debt is emphasized in the following Hadith:
Sahih Muslim: All the sins of a Shahid (martyr) are forgiven
Sunan Ibn Majah: The Messenger of Allah said: "The believer's soul is suspended by his
debt until it is settled for him."
Saheeh Bukhari: A dead person was brought to the Prophet (ﷺ) so
that he might lead the funeral prayer for him ... Another dead person
was brought and he asked, "Is he in debt?" They said, "Yes." He
(refused to lead the prayer and) said, "Lead the prayer of your
friend." Abu Qatada said, "O Allah's Messenger (ﷺ)! I undertake to pay
his debt." Allah's Messenger (ﷺ) then led his funeral prayer.
Wealth is distributed among the heirs in proportions prescribed by Shariah, and the absence of a will is irrelevant. If your mother has no other living heirs (parents) then this would lead to 1/3 going to you, 1/3 for your brother, 1/6 for your first sister and 1/6 for the second sister in accordance with:
Quran 4:11 Allah instructs you concerning your children: for the male,
what is equal to the share of two females ... [These shares are] an obligation [imposed] by Allah. Indeed, Allah is ever Knowing and Wise
If you are thinking of justifying it as a gift rather than a debt, then again it is just to distribute it among the heirs since the Prophet instructed people to be equitable with their children when it comes to gifts:
I heard An-Numan bin Bashir on the pulpit saying, "My father gave me a
gift but Amra bint Rawaha (my mother) said that she would not agree to
it unless he made Allah's Messenger (ﷺ) as a witness to it.
So, my father went to Allah's Messenger (ﷺ) and said, 'I have given a
gift to my son from `Amra bint Rawaha, but she ordered me to make you
as a witness to it, O Allah's Messenger (ﷺ)!'
Allah's Messenger (ﷺ) asked, "Have you given (the like of it) to
everyone of your sons?"
He replied in the negative.
Allah's Messenger (ﷺ) said, "Be afraid of Allah, and be just to your
children." My father then returned and took back his gift.