Are there any fics that focus on Naruto and Neji being teammates/ best friends? I have a weird hankering for a fic that covers Naruto rising up the ninja rankings and Neji replacing Sasuke as his best friend.

 

I wouldn't mind something like that myself. Even with stories where he graduates early, he's usually Neji's replacement. These are the closest things I know of but they don't really fit the bill:

Twenty Matters of Pride to Hyuuga Neji
All Hyuuga are proud. Neji measures his pride differently. First in 'Matters, Secrets, and Times.'

Naruto: Hyuga Pilgrimage
It's time for the Hyūga Heir to make a pilgrimage and only one ninja is up to the task. Uzumaki Naruto, the most unpredictable ninja ever.

Naruto: Eleven
A dark hallway...panic, fear, anger...suddenly she's splashing into a large body of water and turning...."I want her to kill me"..."So you're from Akatsuki"....

 

Huh, hadn't realized Eleven had updated, better read the new chapter. As for the 20 truth one, I'd read it a while ago, probably when I first read 20 times Naruto didn't die. I think I prefer the Neji one to be honest, although both of them are up there in terms of the most interesting stuff I've read on FF.net.
I'd glanced at the Hyuga pilgramige once of twice but the words Hinata and romance put me off, still I'll give it a try.

I have found/ remembered this one: http://www.fanfiction.net/s/4429314/1/Full-Moon
It's not great by any stretch of the imagination, but it is a decent enough fic of middling quality, like most of the author's stories actually.

 

http://www.fanfiction.net/s/8157710/1/Naruto-Iron-Soul
This is the only one that comes to mind for me. Naruto with an iron bloodline replaces Lee.