The blue stone
Eragon finds the stone on a hunt and brings it home. It's an egg, and a blue dragon hatches from it. When he touches her, a silver mark appears on his palm. He raises her in secret, and she picks her own name, Saphira, speaking straight into his mind.
The secret doesn't last. The Ra'zac, the king's servants, come to the farm looking for the egg and kill Garrow. Eragon blames Saphira and sends her away, but soon calls her back, and she forgives him.