Naruto's character would likely become more complex. With the new kekkei genkai, he'd have new responsibilities. He'd need to learn how to use it properly and not let the power go to his head. This would add a new layer to his growth as a ninja.
He might become more introspective. Having a kekkei genkai that others may not understand could make him feel a bit isolated at times. But this would also force him to look inward and understand himself better. He could also use this new power to better understand the struggles of others with unique abilities. For example, he could relate more to Sasuke's power of the Sharingan. This would further develop his empathetic side.