Perhaps I'm jumping in unannounced, but I couldn't help but notice that not many people at the start of this thread are fans of reborn fics, mainly because they were (and perhaps still are, i don't know) wildly popular and generally poorly written. I mostly agree with this. Mostly. One thing I'd like to say is that many of the reborn fics that spawned from the original, which is indeed A Little Night Music over on ff.net, disgust me because they were spur of the moment, "Oh, this is cool, I can do it too!" fics that usually had very similar plots or characters. However, A Little Night Music actually wasn't bad; in fact, I enjoyed it, partially because it was the original and partially for the following reasons.
While riddled with grammatical errors that made me want to vomit at times, the story was solid after the initial few chapters. Not the grammar; that did not improve, unfortunately. The story, feelings, and meanings conveyed developed and improved in quality as the story progressed. By the end, the story had me thinking on an even deeper level than most stories do.
I think its strongest selling point was the character development; while each character did have a depressing, violent, or downright dark past, they were all still varied in personalities, each having a different way of either hiding their past or dealing with their past. I have to admit, though, the excess of horrible pasts became rather vanilla after a while, but it wasn't too bad. And even with a myriad of characters, almost each and every one had some kind of development throughout the story, whether it was the main character realizing the follies of her suicide or the midget Skarmory learning to trust others.
Anyways, maybe that whole portion of text means nothing to you, but this is the point I'm trying to get at. We all have things that just turn us off, but what's not to say that there aren't exceptions? A Little Night Music is (kind of) my exception to disliking reborn fics. I'd like to think that they have great potential if they evolve on their own, away from the ALNM plot and character pool. (This thought actually lead me to begin work on my own reborn fic, and I'd like to think it's coming along well.)
Anyways, what really annoys me in a fic are as follows:
1) Overly generic and uncreative stories: journey fics, reborn stories, etc. There are some exceptions to this, though. I'll read a journey fic or reborn fic as long as they have unique perspectives, characters, plots, or some combination of them all. I.E. a journey fic that is told from a unique perspective and does not follow the games is usually OK, as long as it doesn't fall under my next category.
2) Rampant grammatical errors. I'm not talking a few here and there, I'm talking about all over the place. Sometimes the story and meaning will draw my attention away from this as it did in A Little Night Music, but it usually won't.
3) I agree with Mizan de la Plume Kuro (that's a lot to type, man) and Dagzar. I hate 'chosen one' fics. I wrote one once, and I started loathing it halfway through. Never finished it.
4) Shipping/romantic/anything with over the top love. I enjoy a little happy romance every now and then, but a whole story dedicated to one pairing? Too far, my friend.
5) Using characters from the games/movies/anime/manga in a fic isn't my cup of tea. There are a select few that do well, but most of them just end up annoying me. I never liked many of the anime/manga characters. I also tried writing one of these and after I finished it, I never looked back.
That's mostly what turns me away from a story, but I'd like to think that each one has a couple of exceptions. I'm hoping to create one for the reborn genre with my fic, and I hope I can succeed.
Oh, one more thing: Azurne pointed this out first, but the title of a story is what determines whether I choose one fic over another. An interesting, catchy, and unique title are very important in generating interest. I too avoid most stories with the words 'journey', 'adventures of', etc. Stories with names like a good one I found recently, 'The Thinking Man's Guide to Destroying The World', catch my attention. That's actually how I stumbled upon A Little Night Music; interesting title, so I gave it a shot.