Two other friends and I have been writing a fan fiction and we basically write a chapter each one after another every week. Now that we're nearing the end of the project, we've noticed that the writing style keeps changing and that's it's quite horrible, so what should we do? Should we elect someone to rewrite every chapter so that the writing style stays consistent or we should just roll with it and keep things unchanged for the most part?
I would recommend two rounds of edits, for each author. If the friends are A, B, C (Ariel, Bethany, Cindy) then:
1) First Round: A->B->C->A. Ariel makes edit notes for Bethany, Bethany makes edit notes for Cindy, and Cindy makes edit notes for Ariel. Then each person gets their edit notes, and the author of the chapter makes changes if they want, or gets more clarification from their editor, but the author has the final say.
2) Second Round: Backwards. A->C->B->A. Ariel makes edit notes for Cindy's revised chapters, Cindy makes edit notes for Bethany's revised chapters, Bethany makes edit notes for Ariel's revised chapters.
Again, each author makes changes if they want, or gets more clarification from their editor, but the author has the final say.
3) You are done. Everybody had input on every chapter, but each author retains some level of individual voice in the chapters she wrote, and she decided whether to make changes her way, or rewrite, or whatever. So nobody should feel their voice was shut down or some bully boss (I'm lookin' at you, Cindy!) took over the project and made it her own.
If you are worried about very different voices, introduce three narrators, Debra, Elsi and Fiona for Ariel, Bethany and Cindy, and open the chapter with sub-heading of the narrator's name, "Fiona", or "as told by Elsi", or work the name into the first sentence. Or wrap the story as told by the three women that lived through it. Anything like that.
The first step is to work out some style guidelines among yourselves. Agree on what style you want the finished product to follow. Because this is a project among friends rather than, say, a corporate publication, you'll probably end up including aspects of each writer's style while moving the whole thing toward a compromise center.
Once you agree on what the style rules are, don't each go revise your own contributions. Even with good intent, you're naturally going to favor your own style because you're so used to it. You'll miss things. Instead, revise each others' parts. You might need to do this more than once. When you think you've mostly converged, look for a beta reader from outside your group and see what that person notices.
My documentation team has used peer editing to good effect. We already have a thorough style guide and everybody intends to follow it, but there's drift. We get better results when a second person makes a pass through the work.