In this kind of fanfiction, Luke might have an emotional reunion with his family. He could finally have a long talk with his father, Anakin, about the past and redemption. Maybe he'd share his adventures with his mother, Padmé, if she were somehow brought back in the fanfic's alternate reality.
One possibility is that Luke's visit to his family in the fanfiction would be filled with revelations. His family could tell him secrets about the Force that had been passed down through generations but lost over time. There could also be a celebration, like a family feast, where Luke gets to see his family members interact in a more normal, non - war - torn setting. This would give him a sense of peace and belonging that he may not have fully felt before, as he has been so caught up in the galactic struggle.