The thing that fascinates me about those stories is the sheer amount of construction they have to do. Lucius in canon is a cowardly, bigoted bully, but he's also got that aristocratic flair and a deeply entrenched family loyalty. Reader-inserts provide the scaffolding to chip away at that over time, usually by placing the reader as an outsider—maybe a muggle-born in the Ministry, a Healer forced to treat him after the war, or even a foreign diplomat. The growth feels almost archaeological; you start with his polished, hateful exterior and have to unearth the man underneath all the pureblood dogma.
It's rarely a swift redemption. A good fic will have him cling to his prejudices, using them as a shield, and the reader character has to challenge him not with grand speeches but through small, persistent acts that defy his worldview. His growth is measured in reluctant concessions, a slowly dawning horror at what he's supported, and a redefined sense of honor. The most satisfying ones end with him making a choice that would have appalled his younger self, not for grand forgiveness, but because his new loyalties quietly eclipsed the old ones.
I've seen some fantastic ones where his post-Azkaban fragility is the starting point, and the reader becomes an anchor. That vulnerability, so absent from his early years, is a rich soil for change.