The meaning of ʿUmar’s statement, “We used to seek Your favor through our Prophet ﷺ, and now we seek it through the uncle of our Prophet,” is this: in the Prophet’s lifetime we would go to him, ask him to pray for us, and draw near to Allah by means of his supplication. Now that he ﷺ has joined the Highest Companion and can no longer supplicate for us, we turn to the Prophet’s uncle al-ʿAbbās and ask him to pray on our behalf. It most certainly does not mean that they would say in their prayers, “O Allah, by the rank of Your Prophet, grant us rain,” and then, after his death ﷺ, begin saying, “O Allah, by the rank of al-ʿAbbās, grant us rain.” Such wording is an invented form of prayer that has no foundation in the Book or the Sunnah, and none of the righteous predecessors—may Allah be pleased with them—ever practiced it, as will be discussed in more detail shortly, Allah willing.
The same principle is illustrated by what the great scholar Ibn ʿAsākir—may Allah have mercy on him—recorded in his History (18/151/1) with a sound chain⦗1⦘ from the distinguished Successor Sulaim ibn ʿĀmir al-Khubaʾirī: The sky withheld rain, so Muʿāwiyah ibn Abī Sufyān went out with the people of Damascus to perform the rain-prayer. When Muʿāwiyah sat upon the pulpit he said, “Where is Yazīd ibn al-Aswad al-Jurashī?” The people called for him, and he made his way through the crowd. Muʿāwiyah ordered him to ascend the pulpit and sit at his feet. Then Muʿāwiyah said, “O Allah, today we seek intercession with You through the best and most virtuous among us. O Allah, we seek intercession with You today through Yazīd ibn al-Aswad al-Jurashī. Yazīd, raise your hands to Allah.” Yazīd raised his hands, and the people raised theirs. It was not long before a cloud rose in the west like a shield; a wind drove it forward, and it poured down upon us until the people could scarcely reach their homes.