A possible way for a DBZ and Evangelion crossover fanfiction to work is in the realm of character interactions. Consider Vegeta and Shinji. Vegeta is all about pride and power, while Shinji is often insecure. Their meeting could be a clash of personalities. Vegeta might look down on Shinji at first, but then as they face a common enemy, like an Angel from Evangelion that has some sort of connection to a DBZ villain, they start to understand each other. Also, the technology in Evangelion could be something that the Z - Fighters are interested in. For example, the entry plug system could be something they study to improve their own space travel or combat capabilities.
One idea could be an Evangelion and Gundam crossover. Picture the Eva units in a battle alongside Gundam mechas against a common enemy. Another could be a crossover with Neon Genesis Evangelion and Sailor Moon. The pilots of the Evas could meet the Sailor Scouts and combine their powers to fight a new threat. And a third idea might be an Evangelion and Pokémon crossover, where the Angels are somehow related to powerful Pokémon and the Eva pilots have to use Pokémon in addition to their Evas to save the day.
It could be about Godzilla emerging in the world of Evangelion. Maybe the Angels in Evangelion are confused by this new giant monster. The Eva pilots might have to deal with both Angels and Godzilla at the same time.
One challenge is staying true to the original 'Neon Genesis Evangelion' lore. Since it has a very complex and deep world - building, any crossover has to respect that. For example, the nature of the Angels and how the Evas work are crucial elements that can't be easily changed.
First, you need to know both the Evangelion universe and the other universe you want to cross it with really well. For example, if it's a Evangelion and Star Wars crossover, study the lore, characters, and technology of both. Then, think of a plot that makes sense for the two universes to collide. It could be a shared threat or a quest that involves both sets of characters. Start with a simple outline of the beginning, middle, and end of your story.
Definitely the Eva pilots, Shinji, Asuka, and Rei. They are the main characters in Evangelion and would be at the forefront of dealing with Godzilla's appearance in their world.
One plot idea could be that the Dragon Balls are accidentally brought into the Evangelion universe. This causes a stir as different factions in the Evangelion world, like NERV and SEELE, try to get their hands on them for their own purposes. Meanwhile, the Z - Fighters follow the Dragon Balls and end up in this strange new world. Another idea is that an Eva unit gains the ability to use ki, like a Saiyan. This Eva then goes rogue and starts causing chaos, and it's up to both the Evangelion pilots and the Z - Fighters to stop it.
A possible plot idea is that Godzilla is drawn to the Geo - Front in Evangelion. The reason could be the powerful energy sources there. The NERV organization is then forced to either work with Godzilla or find a way to get rid of it. They might study Godzilla's biology to see if they can use any of its abilities to enhance the Evas. Another plot could be that an Angel takes on the form of Godzilla to deceive the Eva pilots. This would make the pilots question their usual tactics when facing an 'Angel' as Godzilla has different strengths and weaknesses compared to the normal Angels. Also, a plot where Godzilla's presence causes a shift in the power dynamics between the different factions in Evangelion, like SEELE and NERV, could be really interesting.