What is the legal ruling regarding a woman whose infant child slept beside her, and then she found him dead upon waking, and she suspects that she was the cause of death?

The default is that one's liability is clear, and nothing becomes obligatory merely based on doubt. However, if you are certain that you caused the child's death, such as finding him under you, or if your hand was on his face, or you placed a cover over his face, and so on, then it is considered accidental killing, and you are liable for what is incumbent upon someone who commits accidental killing.

Killing in the sense of error is of two types: direct and indirect. A sleeping person who rolls over onto another and kills him – this killing is considered accidental, as it occurred without intent. Its rulings then apply: the obligation of expiation (kaffarah) and blood money (diyah), and deprivation of inheritance and bequest.

This is based on His (Allah's) saying: "And whoever kills a believer by mistake – then freeing a believing slave and a compensation (diyah) presented to his family, unless they remit it as charity."

The blood money is mitigated for the 'Aqilah (the killer's kin) and is waived if the heirs pardon it. The expiation is freeing a believing slave, and if you are unable, then fasting for two consecutive months. The expiation is not waived even if the blood avengers pardon the blood money, because it is a right of Allah Almighty.