One popular storyline is the boys getting into a huge misunderstanding but their friendship prevails in the end. For example, there could be a situation where they are wrongly accused of something and they have to work together to clear their names, which strengthens their bond.
There might be a fanfiction where a fan wins a contest to style one of the Big Time Rush member's hair. The fan is super nervous but has this amazing vision for a haircut. It could be a story full of anticipation as the fan tries to execute their idea. There could be some obstacles like not having the right tools at first, but in the end, they create a masterpiece haircut that gets a lot of attention from the media and other fans.
Well, in Big Time Rush fanfiction, James often takes on different roles. Sometimes he's the hero saving the day for the band. He might be involved in wild adventures that are not shown in the original show, like a secret mission to find a rare music instrument that could boost their career to new heights.
In a Big Time Rush overprotective brother fanfiction, the brother could be overprotective because of a past incident. Maybe his sibling got hurt during a small concert performance once. So, he hovers around, making sure everything from the stage setup to the security around his sibling is perfect. This could create some tension within the group as the others might think he's being too much, but in the end, they come to understand his motives.
A drastic haircut in fanfiction can make a character seem more confident. For example, if a usually shy member of Big Time Rush gets a bold new haircut, it might give them the boost they need to be more outgoing on stage. They start to carry themselves differently and interact with fans more freely.
A really interesting fanfiction could be set in an alternate universe where Logan and Kendall are not part of Big Time Rush. They could be normal high school students who meet by chance. Logan is the shy nerd and Kendall is the popular jock. They initially don't get along but gradually form an unlikely friendship as they discover they have more in common than they thought.
One possible story could be that Carlos gets injured during a Big Time Rush concert rehearsal. Maybe he falls off the stage while attempting a new stunt. His bandmates are super worried. Kendall immediately calls for an ambulance and James and Logan try to keep Carlos conscious until help arrives. In the hospital, they all wait anxiously in the waiting room, praying that Carlos will be okay.
Big Time Rush is purely fictional. It's designed to be fun and engaging with made-up characters and plots. The creators used their imagination to come up with the concept.
You should also consider their goals and dreams as part of the friendship. Maybe they all have different dreams but they support each other to achieve them. For example, one wants to be a great songwriter and the others help him by giving feedback and motivation. Start with a basic outline and then expand on it as you write.
Maybe they get transported to a parallel universe where Big Time Rush is the ruling musical group and they have to figure out how to get back to their own universe while also dealing with the power-hungry versions of themselves in this new world. It would be full of drama and self - discovery.