Another potential rival could be a young, prodigious martial artist from a new school of martial arts on Earth. This character could be jealous of Goku Jr's fame and constantly try to outdo him in battles and in the eyes of the public. He might use unorthodox techniques that Goku Jr has to quickly figure out how to counter.
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.
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.
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.
They likely have a loving relationship. As Saiyans, they would have a bond based on their race and their shared goals, perhaps related to the Saiyan pride and survival.
Well, Goku is a Saiyan, and Saiyans have a strong sense of family deep down, even if they don't always show it. When he meets his parents in this fanfiction, he could initially be in disbelief. But once he realizes it's real, he would embrace them wholeheartedly. He may even start to show off his strength and skills a bit, like a child showing off to his parents. And he would be excited to hear stories about their Saiyan race and how they ended up in the situations they did.
One idea could be Goku Jr facing a new, powerful enemy that has a unique power related to time manipulation. This would force him to train in ways he never has before and seek help from unexpected allies, like some of the older, wiser characters from the original Dragon Ball Z series.
A plot idea could be that Goku Jr. discovers a mysterious artifact that has a connection to the Dragon Balls. As he tries to figure out its purpose, he gets caught up in a race against an evil group who also wants the artifact for their own malicious plans. This could lead to him making new allies and having to use his wits as well as his strength. He might also have to face some personal challenges, like doubting his own worthiness to be a hero, similar to how Goku in the original series sometimes had self - doubts.