What is the ruling of Islamic law regarding a man registering his son, born to a woman who passed away and had no official documents, under the name of his second wife after her consent, due to his need for civil papers?

If the marriage fulfills the Sharia (definition) conditions, it is valid even if not officially registered or if the wife does not possess identification documents. As for registering the deceased woman's son in the name of the second wife, the basic principle is that it is impermissible due to the falsehood involved, unless it is definitively necessary to avert harm from the son. In such a case, it must be witnessed that he is not her biological son to avoid confusion regarding inheritance and other matters.