The number of people who are actively supporting this change and seeking solutions suggests that this offense is not an invention. Using historically inappropriate language is uncomfortable to me and others.
I never have had any ill intentions when using 'lynch' in this game, and I suspect no player here did either. But that doesn't mean a less-charged word is a bad goal.
Plenty of things are uncomfortable to people - bad words, differing opinions, being falsely accused in a game of mafia, etc. It's certainly a valid motivation for change, but you must admit that being uncomfortable is a rather small point to make.
While this may never become an official rule, it will nonetheless become an unwritten rule and something to judge others by. That is, this community will now associate "eliminate" as an anti-racist signal, and view people who say "lynch" in a negative, racist light. In the outside world, there are plenty of actions which are perfectly legal, but cannot be done due to unwritten societal rules. For example, one cannot get a facial tattoo, fly a communist flag, or say a transphobic slur without risking one's career and social status.
The irony here is that "lynch" is an uncomfortable word in part due to the fear of being judged as a racist. That when explaining mafia to someone else, you feel awkward not because you're harming a black person, but because you don't want the other person to think you're bigoted and backwards.
Anyway, I say all of this stubbornly. If renaming "lynch" actually does increase our inclusivity and reduce racism, I welcome the change, but I am dubious if it will accomplish that and fearful that changes such as these are harmful in the long run.
Interesting related article:
http://www.paulgraham.com/say.html