Is what the mother did—preferring one of her children in material giving over the other, with the other son later dying by suicide—considered a sin and impermissible in Islamic law?

A father must be equitable among his children in giving gifts, based on the Prophet's (peace be upon him) saying: "Fear Allah and be just among your children." It is permissible to favor some of them for a reason, such as need, illness, or knowledge, or to withhold a gift from a disobedient or an innovator. For one who strove and whose striving led him to favor some of his children, forgiveness is hoped for, and he should repent and seek forgiveness.