What is the origin of the forty-day memorial, and is there any legitimate religious evidence for the permissibility of eulogizing?

Eulogizing the deceased is an abominable Pharaonic custom with no basis in Islam. It is a rejection of the Prophet's (peace and blessings be upon him) saying: "Whoever introduces into this matter of ours that which is not from it, it is rejected." Eulogizing the deceased and lamenting him by gathering for that purpose and exaggerating in his praise is not permissible, due to the Prophet's (peace and blessings be upon him) prohibition of lamentations, and because it renews grief. However, merely praising him when he is mentioned, or when his funeral passes by, or to introduce him by mentioning his great deeds, is permissible.