The plot could take an exciting turn. If the companions are on a quest for something, River could either be a great help or a major hindrance. She might know a secret shortcut or she could lead them into a trap. It all depends on how the writer wants to develop the story, but it's bound to be full of twists and turns.
Well, you can expect some great character interactions. River is such an interesting character, and when she meets the companions, there'll surely be some witty exchanges. For instance, she might make a sarcastic comment that leaves the companions both amused and a bit flustered.
You might see the story explore different themes. Maybe it's about trust, as the companions have to decide whether to trust River or not. It could also be about the power of knowledge, since River often has a lot of knowledge that others don't. The fanfiction could show how the companions react to her knowledge and how it affects their journey.