Many authors use dialogue to establish a narrator’s name. In Raymond Carver’s short story “Whoever Was Using This Bed,” a couple wakes in the middle of the night to the ringing phone:
“Answer that!” my wife screams from the bedroom. “What in God’s name do they want, Jack? I can’t take it any more.”
F. Scott Fitzgerald introduces the narrator’s name in dialogue in his novel The Great Gatsby.
Tom Buchanan who had been hovering restlessly about the room stopped and rested his hand on my shoulder.
What you doing, Nick?”
“I'm a bond man.”
Of course, you can always be more direct. Herman Melville famously begins Moby Dick this way:
Call me Ishmael.
Not all first person narrators are named. The narrator in Raymond Carver’s short story “Cathedral” remains unnamed and characterization doesn’t suffer at all.