Well, it could be about Goku Jr learning some of Goku's most powerful moves. Maybe Goku teaches him the Kamehameha. They might also go on an adventure together, like fighting a new and powerful enemy that neither could face alone.
Maybe an old enemy of Goku's who has somehow managed to return. This enemy has heard about Goku Jr and figures that by attacking him, he can get revenge on Goku. This could be someone like Frieza who has found a new way to be resurrected and is more powerful than ever.
One idea could be to focus on Bardock's perspective before his fateful battle. Maybe explore his thoughts about his son Goku being sent to Earth and his own impending doom. He could have some premonitions about Goku's future role in saving the universe.
To develop Goku Jr's character further, he could be given more complex emotions. For example, showing his fear in the face of truly overwhelming enemies, rather than always being fearless like Goku. This would make him more relatable. Also, exploring his relationships in more depth, like his friendship with Pan. Maybe they have a falling out over different views on how to protect their world, which forces Goku Jr. to grow and understand different perspectives.
Another possible main character could be the villain that forces the two Gokus to meet. This villain could be a new, super - powerful threat that the two Gokus need to deal with. Their meeting becomes a crucial part of the story to defeat this powerful foe.
Well, in the 'king piccolo meets his son' fanfiction, it could be that the son has some sort of special ability that King Piccolo wants to use for his own evil purposes. However, the son resists. There could also be a sub - plot where other characters from the original story get involved. For example, some of the good guys might try to protect the son from King Piccolo's clutches. And in the end, the son might either change his father for the better or have to defeat him to save the world.
Definitely include Bardock's last stand. His heroic yet doomed battle is a crucial part of the Saiyan lore. It could be written in great detail, with Bardock using his full power against Frieza's forces.
There could also be a moment of self - discovery for both of them. Goku might see his past self in Kid Goku and realize how far he has come, while Kid Goku could get a glimpse of his future potential through meeting his adult self. They could share stories about their friends and enemies, like how Goku could talk about Vegeta and Frieza to Kid Goku, and Kid Goku could tell Goku about his early battles with the Red Ribbon Army.