It will probably not surprise those of you who know me from IRL that I lean more toward the leave-it-as-is camp, and am somewhat skeptical of efforts to change language to better reflect newly-discovered sensitivities in general. That said, it seems clearly to be the case that most people would rather use a different word, and there's really no reason to not do what most people want. Like, this community doesn't exist for its own sake, it exists to please the people who participate in it. So if changing the word pleases an overwhelming number of people, we should do that. I would potentially recommend conducting some kind of anonymous poll, though, to make sure there aren't a large number of people with strong feelings that aren't voicing them here, for whatever reason.
I guess that's just if you're going to have one new standard word—obviously we can, and perhaps should, simply leave it to the discretion of individual game moderators. I do think the universality of some of our key terms—lynch, nightkill, shot, wagon—is extremely useful and it's sort of unfortunate they are tied up in violent imagery that is potentially offensive!
On that subject, I can understand why "lynch" is specifically a bridge too far. The "theme" of mafia is definitely suspicious and paranoid villagers wrongfully killing people, though. You'd really have to sanitize virtually all the flavor to make that no longer the case, to a degree that I find somewhat alarming. I don't think the fact that the town is executing people, and the scum are shooting them, should really be taken as some kind of problematic endorsement of violence. That starts to go into a "violent video games / comics are bad because violence is bad" "won't someone think of the children" kind of moral panic that I really do oppose. I mean, there's a Pillage card in Dominion, but it would be quite crazy to take this as some kind of endorsement or encouragement of pillaging. I do not think Donald X is soft on the issue of wartime theft and property destruction. (I also would agree in general with faust's point that state-sanctioned executions are not necessarily morally superior.)
Again, I'm not trying to say, "Well we can't stop using 'lynch' because where does that end, slippery slope," because we can in fact just end it after that! So I would be for doing that.
As for a new word, the most important thing is to have something short—we use this word a lot—that can be a noun and a verb, and differentiated from the nightkill. Exile is not bad, though mis-exile is sort of an inelegant construction. The best I can come up with that's not mentioned yet is the word fire. I.e. "the town fired ashersky, ashersky was misfired, you were on the fire wagon." It even has the subtle burned-at-the-stake thing going for it, which again I think is a totally permissible part of mafia's theme.
Just some thoughts!