In some fanfics, Naruto can be made stronger than Goku. Fanfic writers have the freedom to manipulate the power levels and abilities of characters. They can give Naruto new and overpowered abilities that surpass Goku's. However, in the original canon of their respective series, Goku has extremely high power levels and abilities that are on a different scale.
To develop Hashirama's ghost, we could give him a sense of purpose. He could be a protector of the village in the afterlife. So, when he senses a great danger approaching the village, he decides to reach out to Naruto. He could have a deep connection to the natural world even as a ghost, and his powers could be related to manipulating nature spirits. He could teach Naruto how to communicate with these spirits and gain their assistance in times of need.
Fanfiction.net is also a good place to search. There are many Naruto fanfictions there. However, the search functionality might not be as refined as AO3. You may need to dig a bit deeper. Browse through the Naruto category and look for tags related to gay relationships between Naruto and Hashirama or just skim through the stories to find what you're looking for.
One thing could be unique chakra abilities. If in a fanfic, Naruto discovers a new form of chakra that gives him unlimited power, he could be stronger. For example, a chakra that can nullify all energy - based attacks like Goku's ki blasts.
I think it has the potential to be popular. In fanfiction, unique family relations like Naruto being the grandson of Madara and Hashirama can open up a plethora of new plotlines. For example, it could change Naruto's relationship with Sasuke, as their clans' history would be even more intertwined. Also, Naruto might inherit some special abilities from his supposed grandfathers, which is always exciting for fans to explore.
Well, the plot could revolve around the descendant's identity being a secret at first. They grow up thinking they are an ordinary ninja. But then, strange things start to happen. They have an uncanny ability to control plants like Hashirama. This leads to a journey of self - discovery where they find out their true lineage and must then decide how to use their heritage for the good of the ninja world.
One popular 'Naruto Hashirama Heir' fanfic might be 'The Heir's Journey'. It often explores the unique challenges the heir faces in a world filled with ninja politics and power struggles. The story usually delves into how the heir tries to live up to Hashirama's legacy while also forging their own path. Another could be 'Hashirama's Heir: A New Era' which might focus on the heir bringing new ideas and changes to the Hidden Leaf Village, much like Hashirama did in his time.
He would probably use them to strengthen the village. Remember, Hashirama was the founder of the Hidden Leaf. His memories could hold secrets about how to fortify the village defenses. Naruto could use that knowledge to make the village safer from external threats.