What is the ruling on saying "May God's name be upon you," or "May God's name be upon your heart," or "May God's name be upon your soul?"

There is no harm in saying, "May Allah's name be upon you," or "May Allah's name be upon your heart." This is a form of protection and seeking blessings through the mention of Allah's name. Similar phrases have been reported in weak hadiths, such as: "In the name of Allah upon myself, in the name of Allah upon my family and my wealth," and "In the name of Allah upon myself, my wealth, and my religion." These can be used as supporting evidence for the permissibility of such expressions.