Roran: There is a possibility that Roran might become the new rider. Because of Eragon's dream in the first book, Roran's dragon would fly off with Saphira when Eragon leaves Alagesia forever. Considering the title, Inheritance, and the fact that all the new Riders are seemingly "related". There is also a part in Eldest says that Roran was convinced in his heart of hearts (he doesn't have a heart of heart, but dragons do). Garrow was quoted in Eldest, speaking to Roran, about their land, that "..if you take care of it, it takes care of you. Not too many things are like that." Other than land, a dragon does that. Also, Tamerlein, the green dragon rider sword, could very well become the sword of the green dragon's rider. The sword is described by Eragon as being meant for someone with a fighting style that relied more upon cutting and slashing. This style seems very appropriate for Roran, saying as how Roran prefers to fight with a hammer as of now. However, the brightsteel that Eragon retrieved from the Menoa Tree was enough for a few more swords or other types of weapons. Rhunon could use the same tactic in creating a new sword for the next rider if they had the speed of an elf. If he is the rider than maybe he breaks tradition and has Horst (with Rhunon's help) forge a brightsteel hammer. It also could possibly be Roran because in the third book he said that he wanted to learn magic.

Arya: It is quite possible that Arya will be the next rider. It is likely the Rider will be female considering that it is suspected the green dragon will be Saphira's mate, and if Saphira falls in love with the green dragon, Eragon's affection will most likely transfer to the green dragon's Rider. Because of the bond between Eragon and Saphira and also because of his dream where he was leaving on a ship with a mate and over the ship were two dragons flying side by side in the first book. Eragon has been infatuated with Arya in the past. Although this led to a confrontation between them in Eldest, the two of them worked together and bonded in Brisingr. Arya revealed her past and her conflicts to Eragon, and the two appear to be getting closer, another incident of this is on their way back to the Varden when she heals Eragon's hand when he was clearly capable. She also looks to Eragon for comfort when Oromis and Glaedr die. In addition, her magic and eyes are as green as the color of the supposed dragon. Paolini states that when someone, elf or human, bonds to a dragon, their magic automatically changes to the dragon's color, regardless what it might have previously been. Arya is the most likely candidate because of the events that took place in Brisingr. Eragon and Arya fighting together against the Shade, connecting their minds when they fought Murtagh and Thorn, and also the previously stated emotional connection and influence. Arya would be a convenient candidate. Also it is unlikely that a human would become the rider due to the fact that they must be able to fight against Galbatorix. Arya already is a great swordsman and is adept in magic, thus she needs little training (except for the green dragon, which would receive training from Saphira). Also, her magic is green and the dragons and its rider are suppose to have the same color of magic.
Nasuada: Nasuada is a strong possibility for the next rider. She is likely because she is female, and also because she would be one of the first people to come into contact with the dragon egg before Arya if the Varden stole it.
i realy dont want nuesuada or roran to be the next rider i would prefer if it were arya than it would all be perfect. and i really hope that eragon adn saphira get to stay in alegasia forever lol
Ragan

Really? I think I read it somewhere........ Or someone here told me.......
It's major role not hatching, that would be a bit of a let-down though.....