DS9 "Sons of Mogh":
KURN: I have never understood you, Worf. But I do know this. In your own way you are an honourable man. (Kurn passes out.)
WORF: And you will be an honourable man again, but not as my brother.
Then right after:
WORF: What will he remember?
BASHIR: He'll know he's a Klingon, and how to speak the language, and virtually everything he needs to know in order to survive, except who he is. And that's the first question he'll ask. Who am I? Do you have an answer?
WORF: There is a man named Noggra, a friend of our father's. He will be arriving in a few hours. He has agreed to provide Kurn with a new identity and a new family. He will supply Kurn with all the answers he needs.
BASHIR: Are you absolutely certain about this, Worf? Once I've erase his memory engrams it'll be almost impossible to restore them. He won't remember you or anything about his real life.
WORF: It is the only way. You may begin.
Kurn's lineage is still the house of Mogh a dishonored house. Kurn's "soul" is still the same. He wanted Worf to kill him to regain his honor so his soul would go to Stovokor (Klingon Heaven).
Erasing Kurn's memory, changing some DNA, doing plastic surgery, and changing where he lives hardly seems to change the main issues which caused his dishonor. His lineage is still the same. If simply adoption into a new house was a legal way to regain honor they could have done this with Kurn's consent.
How does this restore Kurns honor?