If I remember correctly, the book does not say anything about why Saphira as an egg chose Eragon as her Rider. Or did I miss out something? Can anyone help me understand why Saphira chose Eragon?
Saphira chose Eragon, because eggs choose riders that they feel will suit them the best. That is why Galbatorix (when his dragon was killed) had to link another dragon to him by black magic. Dragons pick one person and a person can only be picked once.
The Riders then cast a spell on the egg, causing it to hatch only when the dragon inside chose a human or elf who had touched it that they deemed worthy of becoming partner of their heart and mind. If the dragon accepted the person as his or her Rider, the egg would then hatch. Afterward, the person who first made contact with the dragon hatchling (ideally, the one the baby dragon chose as his/her Rider) would be connected to the dragon forever, linked in both the mind and the body. The person had no choice; the dragon and human were connected until the death of one in the partnership.
This special connection was symbolized in two ways: through telepathy and a Gedwëy Ignasia, a shiny silver mark on the palm of the Rider where the magic originates from.
After the dragon Saphira hatched for him, Eragon became the first Dragon Rider to be born in a hundred years and was subsequently trained by Brom and later Oromis.
From the fact that Galbatorix had to search everywhere for a dragon rider means that riders have to be picked by the dragon, but it would seem random to humans as they don't see that the dragon is thinking and judging.
My theory: Maybe they have to have a parent that was a rider or a relative. I believe in the first canon theory, but genetics may play a part.
Galbatorix tried desperately to find a Rider for whom the eggs would hatch, but was unsuccessful. None of the dragons responded to the touch of anyone in the Empire.
Yet another theory:
You could say that he was chosen due to this person being his namesake. And that it was influenced that he would be named Eragon after the first Eragon who did a similar task and lived a similar life. But then someone could also that that does not count as that would have been influenced by someone else and not Saphira, possibly Angela.
I think that all these theories mixed into one are the reasons why Saphira chose Eragon.
1. The dragon chooses.
Saphira chose Eragon because of his heart. Dragons will wait millions of years to find someone they want to hatch for. When people touch a dragon egg, the dragon looks mainly into the person's heart, so Saphira chose Eragon because he has a good heart.