Perhaps they would explore their family secrets. Since Naruto has always been in the dark about some aspects of his family, with just the two of them, they could start looking into old family mementos or stories that might lead to uncovering hidden truths.
They would probably start by catching up. Naruto has a lot of adventures, and his sister would be eager to hear about them. They would talk and laugh, reminiscing about old times.
In this fanfiction, Naruto may find himself in a situation where he's being hounded by people for various reasons - some for help, some for his power. He simply wants to be left alone to live a normal life. So he might start to avoid the village and the people in it, seeking refuge in a secluded place. There, he could explore his own thoughts and feelings more deeply, perhaps even learning new skills or abilities on his own, away from the watchful eyes of his friends and the village.
In the Naruto series, Naruto doesn't have a biological sister in the original story. So in this fanfiction, the author probably creates a new character to be his sister. It could be an original character with unique abilities and a complex backstory that leads to the situation where Naruto dies by her hand.
He might meet an old sage who has been living in seclusion near the village. The sage could see Naruto's loneliness on his birthday and offer him some ancient ninja knowledge.
She could be an original character created specifically for this fanfiction. The author might give her unique abilities and a distinct personality to make the story more interesting.
In this fanfiction, Naruto could be visited by the spirit of his father, Minato. Minato could offer some words of wisdom and comfort to Naruto on his birthday. They might have a heart - to - heart conversation about Naruto's growth as a ninja and his future in the village.
It could be someone from the village who doesn't understand Naruto's need for solitude. Maybe a council member who thinks Naruto is shirking his duties and tries to force him back into the spotlight.
One possible reason could be to protect her. Naruto might believe that by leaving, he can draw danger away from his sister. For example, if there are powerful enemies targeting him, he doesn't want his sister to get caught in the crossfire.
The relationship could start with Naruto being overly protective of his sister, but she wants more independence. This causes friction. Later, when she gets into trouble that only Naruto can help her with, he steps in. This makes her realize that his over - protectiveness comes from love, and they begin to repair their relationship. For instance, if she is kidnapped by a powerful enemy, Naruto uses all his skills to rescue her, and during this process, they have heart - to - heart conversations.