I wouldn’t even put it on the board. It’s just too obscure to be worth it.
The Kennedy assassination was a few years before I was born, but I knew immediately that this was a
Thursday [editing my derp]
Friday. First of all, it's not obscure to those who lived through it, just the same as those of us who remember 9/11 know that it was a Tuesday. (Maybe I'm assuming that is common knowledge, though.)
Part of my learning of the event was that it was a pall over the weekend, and the oft-told story of how the NFL forged ahead with its schedule, even though the games would not be televised due to unbroken continuous news coverage. I also specifically remember hearing the Oswald murder being on Day 3 and it was a Sunday.
Again, to draw in the NFL as the marker, one of the reasons I remember the Pearl Harbor attack was on a Sunday is that the military personnel in attendance for the Washington football team were paged to return to their command post. Weird that football factors into two events like that.
Beyond those three dates, I don't know if there is any other unscheduled national event that anyone would reasonably know the day of the week for.
tl;dr, I don't think the day of the week of the JFK assassination is an obscure question, but it might have a generational sliding scale of difficulty. (I also thought the president who appeared on Laugh-In was a slam dunk for a $1 million question, so maybe it's just the way my brain is wired.)