You must be at least 10th level to activate this quest. At some point after delivering the Unicorn Horn to Medora, she will contact you and tell you to go see Gortwog about the Dust of Restful Death, which is supposed to put Lysandus' spirit to rest.
This quest may also be accidentally activated if you try to do Elysana's second quest (The Setup) and Gortwog is the person you're supposed to see. It seems that if you try to talk to Gortwog at any time after talking to Medora for the first time, it activates this quest prematurely.
Gortwog will tell you that the dust will only calm Lysandus, not put him out of his misery. In return for his help in telling you where to locate the Dust, Gortwog wants Medora's support for his claim to the "heart of Tiber Septim" (the magic dingus you're supposed to be finding).
The Dust will be in a random dungeon in a random location within the dungeon. The key is that you're not looking for the component. You're looking for a mummy which will have the dust and a letter on its body. If you're using one of the later patches, the dust and letter will be on a green background (like any other quest object). If you're not using a patch, make sure that the mummy has lich dust AND a letter that says something about Medora. If it doesn't have both, then you haven't killed the right mummy. When you've got it, take it to Medora. She will tell you to come back in a month to get the finished product.