A lot of fics in which the genin teams are different, either because someone else is teaching Team 7 or someone else wanted Naruto (such as, for example, Team 8), which shuffled the teams around. (These fics are most often Naruto/Hinata)
The in/famous Clan Restoration Act, where one male character is forced/ or wants to pair with multiple women to revive his clan.
Also there multiple fics when Naruto gains an awesome power and virtually every girl except (in most cases) Sakura, usually fall for him that includes his mother, Tsunade and girls he never met like Yugito.
Nearly every story starts with Naruto being hated and attacked by the citizens of Konoha and the Third doing absolutely nothing about it or is too late.
He is sometimes saved by a crossover character or an OC.
Also there are many fics that deal with Naruto bringing back Sasuke only to be betrayed by Sakura and is later exiled/executed by the elders and the "civilian council".
It usually ends up with Naruto gaining Combo Platter Powers, becoming the leader of a village while Konoha gets destroyed economically and loses their allies as well. With war between Orochimaru, Kumo and Iwa.
Twelve-plus young characters of the same age, with a variety of personalities? Let's put them in high school!
Another really common one is the non-massacre AU, usually used as an excuse to pair Itachi and Sakura up.
Fanfics where Naruto turns out to be the last survivor of a made up clannote Usually the Namikaze and gets a new bloodline (particularly the Rinnegan) or Kurama some unexplained way gives alters his genes giving him the bloodline. Before the Uzumaki Clan was revealed in canon but Naruto still doesn't have a bloodline.
Wouldn't the fact that all Uzumaki possess pretty much the same physical attributes count as a bloodline? Kushina flat-out says that the Uzumaki were famous for their massive chakra levels, almost endless stamina and inherent skills in fuinjutsu.
Another very popular is NaruHina shippers making Naruto as dumb as possible and having constantly mocked by everyone for not noticing Hinata's feelings. Hinata, herself is usually portrayed as a stuttering like crazy for two seconds before fainting.
Sakura is also portrayed as a screeching harpy and Sasuke's loyal groupie. This has some canonical basis as in the early days Sakura would yell at Naruto for insulting Sasuke, but because a number of fans hate her, this gets taken Up to Eleven in fanfics.
There's also the ever popular de-aged fic, which typically involves a jutsu going wrong and the main character of the story being turned into a child version of him or herself.
A lot of crossovers usually start with the Naruto and Sasuke fighting at Valley of the end and the Kyuubi's and the Curse Seal's chakara will merge and create a portal taking Naruto to a new world.
Usually as an excuse to make Naruto "Better" are plots that go "what if Naruto was 'realistic?note realistic in this case often meaning turning Konoha into a military dictatorship, or some other bizarre alteration to canon'"
The Uzumaki clan turns out to have survived the Iwa and Kumo attack.
Naruto snaps and become a cold-blooded killer.
Crossovers usually make it so outside the Elemental Nations is a part of the world of whichever series the author may choose which will be referred as 'outside the veil'.
A lot of crossovers mostly The DCU have Naruto be a Demon and a cult dedicated to him will awake him and he'll often be a villain.
Superhero crossovers will have Naruto be an antihero who'll kill villainsnote Even though he doesn't believe in killing either and some crossovers may take place after he fought Pain and established his no killing rule and will berate certain heroes for not killing.note Mainly because the author wants them to despite the fact most heroes will usually have grounded reasons for not killing.
Though it's worth noting that quite a few examples use an alternate backstory for Naruto (being trained and joining ANBU, for example) that led to him not developing his canon moral code or have him come really close to crossing the Despair Event Horizon at some point in the series and changing because of that.
Minato and Kushina survives the Kyubi attack and has another child. They later focuses all their attention on the child completely neglectingNaruto in the process..
For some reason the other child will usually be a girl.
There are many fics with Sakura having just about every Naruto guy under the age of 30 fall in love with her and it's pretty popular to top this off with AU as well.
Some will have it that Kurama invented the Sharigan mostly just so Naruto can get it.
The reason Kurama attacked Konoha is someone(usually Orochimaru) wearing Konoha headband killed his (or depending on the story her) cubs.
Not necessarily a huge part of a story but when Naruto meets someone there may be a debate if Naruto's name stands for 'fishcake' or 'maelstrom'.
Danzo kidnaps Naruto and wipes his memories clean turning him into a Wild Child in the process.
Kurama mentors Naruto, calls him "Kit," and helps him learn assorted cool demon powers.
While not always the main plot of a fanfic, some fanfics have Naruto decide Sasuke is beyond redemption and kill him or capture him and leave him to his punishment. These are often done by people who dislike Sasuke and/or the idea of Naruto continuing to try to bring him back after all he's done.