It could also be a situation where neither of them initiates it directly. A third party, like Splinter, could observe the ongoing conflict and see the potential for peace. He could then encourage Leonardo to reach out to Tigerclaw, or somehow manipulate the situation so that a truce becomes a possibility.
Leonardo would be a likely candidate. He is often the most level - headed of the turtles and believes in honor and justice. He might see the potential for a greater good if he can end the hostilities with Tigerclaw, so he could be the one to make the first move towards a truce.
One idea could be a story where Leonardo and Tigerclaw are forced to team up temporarily to defeat a common enemy greater than their rivalry. Maybe a new, powerful mutant emerges that threatens both the turtles and Shredder's plans. This would put them in a situation where they have to rely on each other's skills.
The conflict between good and evil is a key element. Leonardo represents the good, and Tigerclaw is clearly on the side of evil. Their clashes, both physical and ideological, can create a lot of tension and excitement. For instance, their different fighting styles - Leonardo's ninjutsu training with Splinter versus Tigerclaw's more brute - force and underhanded tactics. Another element is the mystery around Tigerclaw. There could be parts of his past that are slowly revealed in the fanfiction, which could add depth to the story.
One possible storyline could be a battle of wits between Leonardo and Tigerclaw. Leonardo, being the leader of the Turtles, would constantly try to outsmart Tigerclaw. For example, he could set up elaborate traps to catch Tigerclaw and stop his evil plans. Tigerclaw, on the other hand, would be constantly trying to anticipate Leonardo's moves and turn them against him. Another storyline could be a more complex relationship where they start to understand each other's motives over time. Maybe there could be a situation where they have to team up against a common, even greater enemy. This could lead to a begrudging respect between them.
It could be Louis. He has always seemed very connected to the group's roots. In fanfiction, he might realize how much the fans miss the band and how much they all still have in common musically. So, he starts reaching out to the other members one by one to see if a reunion is possible.
Goku would be more likely. He is always open to new friendships and challenges. He has a curious nature and would be intrigued by Kefla's power. So he might approach her first to test her strength or just out of friendly curiosity.
It could be Naruto. He's known for his boldness and his ability to reach out to people. He might be the one to first notice Korra's unique qualities and be drawn to her, starting conversations and gradually building a connection that could turn into a romantic one.
The Shredder would be a likely villain. He could see the Hunger Games as an opportunity to finally get rid of the TMNT. He might manipulate the other tributes or even work with the Capitol to set up traps for the Turtles.
Koro - sensei could be a likely candidate. Since he has advanced abilities, he might be able to figure out time travel in some way. He could use it to try to change his own past or to better prepare the students for their future.
Tim Drake from the Batfamily might be likely. He's highly intelligent and sometimes over - thinks situations. If he discovers some unusual activity related to Young Justice that he misinterprets, he could be the one to start the ball rolling on a conflict. For example, if he sees some new Young Justice members using untested tech that he thinks could be dangerous, he might take preemptive action against them.
However, Luke could also be the one to start the conversation. He is a young and idealistic Jedi, full of hope. He might want to reach out to his father, believing that there is still good in him. He could start the conversation with an attempt to understand Vader's journey to the dark side.