Individuals who've laid eggs often keep track of the eggs and resulting kids even if they aren't raising them themselves, and that extra attention means it's a little more likely for the kid to pick that adult to adopt if they don't get along with their first parent, but there's nothing formal in most cases. The exception is chiefs' eggs - chiefs very rarely raise children at all, but it's traditional for them to leave the tribe to one of their descendants when they retire, so there's some extra attention and rules about that.

(Her egg is not a chief's egg - her tribe's chief didn't have an egg, this year, but even if they had she wouldn't've taken it, even if it was the only option. She herself being chief's-kin and a speaker was enough of a fiasco; intentionally interfering with the succession like that is a political morass she has no interest whatsoever in wading into.)