Praise be to Allah.
The basic principle is that she is innocent unless it is proven that the offence was committed and that she was the cause. If it is proven that the girl died because of severe cold, then your mother has to pay the diyah and offer expiation, because of her negligence and causing the death of her daughter, unless the heirs of the girl agree to forego the diyah, in which case it is waived. In the answer to question no. 52809 it is explained that the diyah in the case of accidental killing has to be paid by the ‘aaqilah of the killer (male relatives on the father's side) and not by the killer himself (or herself); in the question referred to there is also a discussion of what is meant by the ‘aaqilah.
If it is not proven that the girl died because of the severe cold, or there is some doubt concerning that and it is not certain, then your mother does not have to do anything, because the basic principle is that she is innocent and an offence cannot be proven on the basis of suspicion.
Ibn Hazm (may Allah have mercy on him) said: If he [the infant] died because of her [the mother’s] action, such as if she put a blanket over his face, then he slept and turned over and died of suffocation, or her arm fell over his mouth or her breast fell on his mouth, or she rolled on top of him without realising, there is no doubt that she killed him by mistake, so she has to offer expiation and the diyah has to be paid by her ‘aaqilah or by the bayt al-maal. If he did not die as a result of her action, then she does not have to do anything or pay any diyah at all.
If she is not sure whether he died as a result of her action or otherwise, then no diyah is to be paid in that case and no expiation is to be offered, because we are certain that she is innocent of killing him, but we are not certain whether he died as a result of her action or not, and wealth cannot be taken except on the basis of certainty.
The expiation is a shar‘i obligation, and shar‘i obligations cannot be proven except with a text or scholarly consensus. It is not permissible to impose a penalty or a fast, or to oblige her ‘aaqilah to pay the diyah on the basis of a false suspicion. And Allah is the source of strength.
End quote from al-Muhalla, 11/15.