It would likely make her more confident. Having Naruto's eyes might give her a sense of his strength and determination. So, she could become more assertive in battles and in her relationships.
Hinata might become more confident in herself. After seeing Naruto's memories of his own struggles and how he overcame them, she could apply the same principles to her own life. For example, if she sees how Naruto faced his insecurities about being a jinchuriki, she could face her own insecurities about her skills as a ninja more bravely. This would lead to a more well - rounded and strong Hinata.
In such fanfiction, Hinata's actions could potentially stunt Naruto's growth. If she hurts him, Naruto, who is often portrayed as sensitive beneath his bravado, might become more guarded. It could lead to him doubting his own self - worth, especially if the hurt comes from someone he cares about or trusts.
His character might develop in terms of his understanding of family. Since Hinata's mom is part of a traditional and complex family like the Hyuga clan, Naruto would have to navigate through their customs and politics to save her. This could make him more empathetic towards family values and the importance of lineage, which is different from his own upbringing as an orphan.
She becomes more confident. With the Tenseigan, Hinata has a new source of power. This newfound strength gives her the confidence to be more assertive, not just in battle but also in her relationships. For example, she might be more outspoken in the ninja council meetings where she used to be quiet.
Her increasing strength also plays a role. In some fanfictions, Hinata trains hard and becomes a powerful ninja. When she uses her skills to protect Naruto or help him in battles, it pleases him. Naruto has always respected strength, and seeing Hinata grow in power makes him proud and happy, and also makes him see her in a new light.
The relationship would likely become more intimate. They'd have more time to just be with each other, sharing their thoughts and feelings without any rush. For example, they could have long conversations at night before going to sleep.
Naruto's character development could be shown through his increased sense of responsibility. He is no longer just thinking about himself or his goal to become Hokage. Saving Hinata from Orochimaru makes him realize the importance of protecting those he cares about, which is a big step in his growth as a ninja and a person.
His friends would be really happy. Shikamaru would probably be like 'Well, it was bound to happen eventually, Naruto.' and offer some practical advice. Choji might bring over some delicious food for Hinata to help with her pregnancy cravings. Kiba would probably joke around with Naruto about how his life is really changing now.
It could make Naruto more introspective. He might question his own worthiness of Hinata's sacrifice. This could lead to him delving deeper into his own beliefs and values. He may also become more empathetic towards others who have lost loved ones, as he now has first - hand experience of that kind of pain. He would grow in maturity, realizing that his actions have consequences not just for himself but for those around him who care about him deeply.
Hinata would develop more leadership qualities. With Naruto gone, she realizes that someone needs to step up and be a symbol of hope for those who still believe in Naruto. This would make her more confident in making decisions for the greater good of the village and its relationship with Naruto when he returns.