Here "name" refers to the reputation of Yahweh. AT: "So that people will continue to honor and respect Yahweh" (See: [[rc://en/ta/man/translate/figs-metonymy]])
The people are filled with fear because of the rain and thunder that Yahweh sent when Samuel prayed. Some people may believe Samuel would use his prayers to harm them. (See: [[rc://en/ta/man/translate/figs-explicit]])