" As Eragon spoke, an idea occurred to him, one that resonated within him too strongly to ignore. He explained it to Saphira, and once again she granted him her permission, although somewhat more reluctantly than before.
Must you? she asked.
Yes.
Then do as you will, but only if she agrees.
When they finished speaking of Vroengard, he looked Arya in the eyes and said, “Would you like to hear my true name? I would like to share it with you.”
The offer seemed to shock her. “No! You shouldn’t tell it to me or anyone else. Especially not when we’re so close to Galbatorix. He might steal it from my mind. Besides, you should only give your true name to…to one whom you trust above all others.”
“I trust you.”
“Eragon, even when we elves exchange our true names, we do not do so until we have known each other for many, many, years. The knowledge they provide is too personal, too intimate, to bandy about, and there is no greater risk than sharing it. When you teach someone your true name, you place everything you are in their hands.”
“I know, but I may never have the chance again. This is the only thing I have to give, and I would give it to you.”
“Eragon, what you are proposing…It is the most precious thing one person can give another.”
“I know.”
A shiver ran through Arya, and then she seemed to withdraw within herself. After a time, she said, “No one has ever offered me such a gift before…I’m honored by your trust, Eragon, and I understand how much this means to you, but no, I must decline. It would be wrong for you to do this and wrong for me to accept just because tomorrow we may be killed or enslaved. Danger is no reason to act foolishly, no matter how great our peril.”
Eragon inclined his head. Her reasons were good reasons, and he would respect her choice. “Very well, as you wish,” he said. "

― Christopher Paolini , Inheritance (The Inheritance Cycle, #4)