Who has the priority in naming the newborn: the paternal grandfather or the parents? And is it considered filial piety to the father after his death to name the grandchild with a common name that he (the father) desired, or with a Sharia-compliant name that the mother of the newborn desires?

The father has the primary right to name the newborn. If the grandfather desires to name his grandchild a specific name, the father should agree to it and waive his right, unless the name is forbidden by Islamic law. This is because it involves obeying and honoring the grandfather. There is no objection if the name desired by the grandfather is less preferred, while the father wishes for a more virtuous name. The important thing is to honor and obey the father, as long as it is within what is considered good and permissible.