In some fanfictions, Naruto might get the scar on his eye during a particularly intense battle with a powerful enemy. Maybe it was a close encounter with a rogue ninja's weapon that managed to leave a mark on his eye area.
Hermione might be the one. With her extensive research skills and vast knowledge of magic, she could have stumbled upon a way to remove the scar. She was always determined to help Harry, and this would be just another of her great contributions.
His scar might make some people more intimidated by him. They could see it as a sign of his dangerous experiences and be a bit afraid to get too close. For example, new acquaintances in the fanfic might be hesitant to approach him at first.
He might try to self - medicate at first. Maybe he takes some experimental drugs he's been working on in his lab. After all, he's always been one to take matters into his own hands.
Well, it definitely slows him down. He can't push himself as hard as he usually does. His stamina is lower, so he might not be able to complete his normal training routines.