A Name of Power is the 7th chapter of Eragon.
On the way home, Roran tells Eragon that a stranger from Therinsford was at Horst's, a man named Dempton who came to have Horst forge him some sockets. He offered Roran a job. Eragon asks if Roran intends to tell Garrow and Roran gives his assent. Eragon wonders why, given how Garrow feels about them going away, and Roran tells him that he accepted the job. Eragon says that they have enough money, but Roran explains that he intends to marry Katrina. He hasn't yet asked her, but will once spring comes when he can raise a house.
Eragon tells him there's too much work on the farm for him to leave, but Roran says that spring is when he'll be needed the most. Eragon wishes him the best of luck, but warns that Garrow might not take it well. They return home and Roran does not yet say anything, but Eragon is sure he soon will.
Eragon goes to see Saphira, aware properly for the first time that she is an equal. Eragon, she comments, and Eragon asks her if that's all she can say. Yes, she replies and Eragon is bemused that she's now developing a sense of humor. He ends up telling her what happened with Roran and begins to rage because he does not want Roran to leave. He begins suggesting names for her, such as Vanilor or Eridor, but as he continues to make suggestions, Saphira just seems to be laughing. He comes to a revelation that he has missed something important - Saphira is female! He begins suggesting female names, but none satisfy her until he asks her about the last name that Brom told him, the one that he muttered, Saphira. He asks her if she's Saphira and he feels her satisfaction. She begins to hum.