While I was using Stack Overflow at work, someone's just spoilt the twelve-hour old Game of Thrones season finale for me with an extremely ill-advised question title over on SFF.SE. Hot Network Questions popped it up right there alongside the answer I was reading about whether C++ strictly "includes" C from a language lawyer perspective. My efforts to entirely avoid Facebook, YouTube, Quora, Twitter and SFF.SE itself throughout today were ultimately, then, in vain. The question's title has since been edited (presumably by a higher-rep user — I didn't look at anything but the page title), but it's too late.
The least invasive change I can think of proposing right now is to allow us to remove certain sites from the Hot Network Questions rotation. So if we're in the middle of Game of Thrones season, I'd probably remove SFF.SE and Movies & TV from my HNQ display. I do find myself visiting lots of interesting questions on those sites through HNQ, but short of making spoilers punishable by death and hoping that word of this new law spreads, I can't think of anything else to suggest.