Well, it could be a very emotional encounter. Padme might see the great potential in Luke and Leia. She could be proud of them and the Jedi paths they might be on. Maybe she would have some advice for them about the Force and how to use it for good.
It could inspire hope. If the news of Padme meeting her children spreads across the galaxy, it would give the rebels more motivation. People would see that there are still connections between the past and the present, and that the fight against the Empire is not just a random struggle but part of a bigger story.
Their interaction would likely be filled with mutual respect. Padme, as a former queen and senator, would recognize Leia's leadership qualities immediately. Leia, in turn, would be in awe of Padme, knowing her story and her role in galactic history. They would probably exchange stories about their families, with Padme sharing memories of Anakin, and Leia talking about Luke and her adoptive parents.
One possibility could be a time - travel plot. Maybe Padme somehow gets a glimpse into the future and meets Leia. They could bond over their shared experiences of being strong female leaders in a galaxy full of challenges. Padme might see in Leia the kind of woman she hoped her daughter would become, and Leia could gain wisdom from Padme's past experiences in the Senate and in the Clone Wars.
I really liked 'The Time - Traveling Skywalkers'. It's a story where all four characters are caught in a time - travel anomaly. They end up in different points in time relative to each other. Padme gets to see a future where her children have grown up strong despite all the hardships. Anakin has to face his past self and tries to guide him away from the dark side. Luke and Leia use their knowledge of the future to try and make the galaxy a better place in the past. It's a really engaging read.
One great storyline could be a 'what if' scenario where Anakin doesn't turn to the Dark Side. In this, he and Padme raise Luke and Leia together. The twins grow up with their father's Jedi training and the family lives in harmony, facing the challenges of the galaxy as a united front.
Here's another idea. Suppose there's a threat to the galaxy that can only be stopped by using a long - lost technique that Padme knew about. Luke and Leia, with the help of Anakin's spirit, embark on a quest to find out about this technique. They face many challenges, including political intrigue and dark side threats, all while coming to terms with their family's complex history.
Luke is often very well - written. Fanfiction authors can explore his journey with the Force in more detail. They can show his growth from a naive farm boy to a powerful Jedi in a more nuanced way.
A third plotline could be set in an alternate universe where Luke and Leia meet under different circumstances. For example, they could meet as children before they were separated. This would completely change the dynamic of their relationship and lead to a whole new set of adventures and character developments.
It could be that in this fanfiction, Padme is somehow brought back to life (through some unique plot device). She and Luke then team up to fight against a new threat in the galaxy. Luke would be in awe of her, having heard so much about her, and together they would use their combined strength and knowledge to protect the galaxy.