What is the ruling of Islamic law concerning the custom of giving a large sum of money (Al-Hafalah) to a woman who is blessed with a newborn, and does it have any basis in Islamic law?

Yes, giving a gift for a newborn upon its birth is originally permissible, because the fundamental principle in dealings is permissibility unless there is evidence to indicate prohibition. It is a common custom among people that fosters affection and love. However, if this custom entails harm, such as burdening the husband, then it is prohibited due to the harm and embarrassment it causes. As for a simple gift, there is no harm in it.