She may be a dragon, but she's one of the most important characters in the book. In fact, you could make the case that, although this book is called Eragon, without a Saphira there'd be no Eragon. Dude would still be in Carvahall, digging turnips with a spoon or whatever it was he was up to. Don't forget that it is Saphira's choice to find Eragon, and that his heroism is only the result of whatever dragon-y insights she had into his character. More than anyone, Saphira is responsible for unleashing the potential in her partner-in-flight.
So yeah, she's a pretty big deal.
First and foremost, Saphira and Eragon are friends. Their friendship forms the basis of their interaction and makes all the other things they do—casting spells, flying loop-the-loops, slicing and dicing Urgals—possible. We understand right from their very first meeting just how special their bond is.
After they first touch, Eragon feels that "[s]omething brushed against his consciousness, like a finger trailing over his skin. […] It was as if an invisible wall surrounding his thoughts had fallen away, and he was now free to reach out with his mind" (5.5). Sounds pretty freeing to be connected in this way. The bond these two share is unbreakable, and forms the foundation upon which their shared power grows and develops. This is like a Dawson and Joey style connection.
Speaking of power, Saphira functions like a sort of giant, blue battery for our Dragon Rider, lending him her strength when she needs a hand. Not only do they fly and fight together, but Saphira can telepathically send Eragon her vibes to increase his abilities, particularly when he uses magic.
Remember the test that those nasty Twins put Eragon through? Well, Saphira tells him that she's got his back: "Don't worry so much [….] There are two of us as well" (56.27). Good thing, too, since "It was only with Saphira's support that Eragon was able to hold his ground" (56.35). With Saphira by his side, Eragon is able to take on all comers. She's the key enabler of his heroic acts.
Want some advice? Get a dragon. No, really. Saphira is not just there to pump up Eragon's hero muscles. She's someone Eragon can turn to for direction when he's lost. After Garrow passes away and the villagers are getting suspicious, what does Saphira say? "The only true guide is your heart" (14.4). Sure, that sounds like something you'd see painted on cheap coffee mug, but watch its effects on Eragon, who says, "What do you want me do…pursue the strangers?" (14.7). This is a sophisticated bit of dragon psychology. Eragon comes to the decision to leave on his own, and is allowed to do that by Saphira.
Saphira's role here is to gently lead Eragon on the right path, and then support him along the way. Lucky for Eragon that he's got such a supportive, wise, and powerful friend. Saphira accounts for at least half of his hero points. On most days, we think it might be more.