Is there anything incumbent upon the mother if her child died after she covered him with a heavy blanket, fearing the cold for him, and she does not know whether she or the blanket was the cause of his death?

If she covered his face with a heavy rug and he died, then she is liable for expiation for accidental killing, and the blood money is incumbent upon her 'aqilah (male agnates). If she did not cover his face, then there is nothing upon her. The expiation is the freeing of a believing slave; if she is unable to do so, then fasting for two consecutive months. The blood money is incumbent upon the 'aqilah of the killer, not upon the killer herself. If he died as a result of her action—meaning she covered his face and he died of suffocation—then she is liable for expiation and blood money. If he did not die as a result of her action, then there is nothing upon her. If she is uncertain whether he died due to her action or not, then there is no blood money and no expiation, because the default is her innocence.