The child would add a new dimension to their relationship. Naruto and Hinata would have to communicate more about the child's needs. This would lead to a deeper understanding of each other. They might also have different opinions on how to raise the child, but through resolving those differences, their relationship would grow. Hinata might become more assertive in expressing her views on parenting, and Naruto would learn to listen more, which is great for their relationship.
They might face challenges in the child's education. In the ninja world, there are specific skills and knowledge that the child needs to learn. Naruto and Hinata have different fighting styles and abilities. Deciding which ones to teach the child first could be a headache. Also, the child might struggle with making friends in the village. Naruto was an outcast as a child, and they'd want to prevent the same for their adopted child. They'd need to be very involved in the child's social life to make sure he or she is accepted.
It could strengthen their relationship. Taking care of a child together requires cooperation and communication. They would have to work as a team to raise the child, which would deepen their bond. They might also discover new things about each other through the process of parenting.
At first, it might be awkward. Kakashi is used to being a mentor figure, and now he's in a position of being 'adopted'. But as time passes, Naruto's positive energy starts to rub off on Kakashi. Kakashi becomes more carefree and starts to enjoy life more, like Naruto does.
The child could create new shared experiences for them. For example, they would celebrate the child's birthdays together, take family vacations, etc. These shared memories would strengthen their bond. Also, seeing each other in a parental role could make them respect and love each other in new ways. Naruto might see Sakura in a more motherly and tender light, and Sakura might admire Naruto's fatherly instincts even more.
The child might introduce new ninja techniques. Combining the power of the Uzumaki and Hyuga clans, they could develop something truly unique. This new technique could change the way battles are fought in the ninja world. For example, if they can merge the Byakugan's perception abilities with Naruto's chakra control, it could create a whole new level of strategic advantage in combat.
In terms of Naruto's skills, the child could also inherit them in a more natural way. Since Naruto has such strong chakra, some of that could be passed down, allowing the child to learn his jutsu more easily. As for Hinata's skills, the child might inherit her grace and precision in combat. Hinata was known for her elegant fighting style, and the child could pick up on that instinctively, especially if they are trained by Hinata at an early age.
Their relationship would probably become more deeply connected. They would have to work together to raise the child, which would mean more communication and understanding between them. Naruto, who values family highly, would be very involved in the child's life, and Konan would likely become more open and affectionate as a mother.
The adopted child could be a muggle - born with a great potential for magic. Maybe their parents were killed by dark wizards who were against muggles in the wizarding world. Harry and Hermione, having both experienced prejudice, would be determined to give this child a chance.
Perhaps the child is the offspring of a Titan. This would make for an interesting story as Percy would have to deal with the potential danger and mistrust from other demigods. But Percy, being Percy, would see the good in the child and adopt them regardless. The child might have unique powers that set them apart from other demigods, like the ability to control time or space.
Naruto would be a very loving and supportive father. He'd probably be a bit overprotective at times because of all the dangerous things he's faced. But he'd also encourage his child to be independent and follow their dreams. He'd be there to teach them ninja techniques and share his own experiences to help them grow.