Obviously, dead players quitting feels and often is extremely detrimental to the Medium. I don't believe that trying to stop people from quitting is a valid solution because you can't do that without catching people who have nothing constructive to add (their Will says it all or they were a Witch or something who has no reason to care about the game's turn-out after they're out), I believe that quit-proofing the game as much as possible so that it can keep going when someone does quit is the real answer.
Thus, for Medium, my proposal is to give them a new night ability. At night, they can choose to visit a dead player. If the visit succeeds, they will receive that player's History, which is effectively an auto-generated Last Will that summarizes the actions they took each night and the results of those actions. For example...
Day 3 - John Proctor was killed last night. We could not find a Last Will. His role was Investigator.
Night 3 - The Medium visits John Proctor successfully. He receives his History, which can be reviewed at any time after the visit.
The text of John Proctor's History may say something to this effect depending on how flowery/in-character you wanna be (maybe have a game option for In-character Histories and Concise Histories). Bolded text would be highlighted white or a relevant color in game.
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In my life I was John Proctor, an Investigator.
On the first night, I investigated William Phips and deduced that they must be a Bodyguard, Godfather, or Arsonist.
On the second night, the night of my death, I attempted to investigate Ann Putnam but was distracted and unable to do so.
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And, for reference, the Concise History might read something like this.
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John Proctor - Investigator
Night 1 - William Phips. Bodyguard/Godfather/Arsonist.
Night 2 - Ann Putnam. I was roleblocked.
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So the Medium gets the information regardless of whether or not John Proctor has quit, so that if he did quit that's as close to irrelevant as possible (though it could still be a detriment if there were multiple quits that night). This can also be used to discover cleaned and forged roles (as this is possible if the cleaned/forged player doesn't quit), though if the target's true role is Mafia or Neutral (or Coven, I guess, I don't have that expansion), the Medium will be told "You were not able to receive this spirit's History.". Note that this message is completely generic, and will also be shown if the Medium chooses a town role and then get stopped from completing the visit by a Witch, Escort, Consort, or whatever else if anything. (If necessary, have it completely override other messages so that it's opaque as to what actually happened. Careful now, Hypnotists)
I think that'd help because it'd give the Medium something constructive to do even if everyone quits, and as a buff I don't think it'd break Medium much if at all, since it's presently kind of an underwhelming role to handle and this ability would not matter at all in a game where nobody quits.