From egg to sky
Saphira's egg was in the hands of the elf Arya, who sent it away before the Shade Durza caught her. It turns up in the woods, where Eragon finds it. Saphira hatches, grows fast and is soon big enough to fly with him on her back. She picks her own name and talks to Eragon telepathically.
From the start, the bond runs both ways. The silver mark on Eragon's hand shows she chose him, and Saphira has little patience for his doubts: she knows the two of them will have to grow up together, and fast, because the king is already after the egg.