One interesting concept could be the unique powers associated with the red rinnegan. Maybe it has powers that are different from the regular rinnegan in terms of combat abilities, like being able to manipulate a new type of energy or having a special form of genjutsu. Another concept might be how the character who possesses it interacts with other characters in the Naruto universe. They could either be feared or revered, and this would change the power dynamics among different ninja clans.
The red rinnegan in the fanfic could introduce an interesting concept of a new level of ninjutsu. Since the rinnegan is already a very powerful dojutsu, the red version might unlock some hidden techniques that were previously unknown. For example, it could have the ability to not only control the six paths but also some other supernatural elements. And it might also change the way the user perceives the chakra network, allowing for more efficient use of chakra or even the ability to steal chakra in a different way.
Well, in 'naruto red rinnegan fanfic', perhaps a cool concept is the origin of the red rinnegan. It could be tied to some ancient and long - forgotten jutsu or a genetic mutation within a particular bloodline. Also, the fanfic might explore how the red rinnegan affects the user's mental state. It could either drive them towards a more extreme sense of justice or perhaps make them more power - hungry than other characters with normal eyes.