A good technical solution would be to allow mods to exclude questions from specific tags from appearing on the HNQ.
Impact would be low, since filtering is not done per user, so it does not have to take each user's preferences into account. This would simply prevent questions within a tag from reaching the HNQ at all.
Users on Stack Foo have learned (the hard way) that subject Bar is poorly understood by laypeople, drawing bad answers and needing protection whenever a question about Bar hits the HNQ.
Mods on Foo.SE add bar to the list of HNQ-banned tags. Questions tagged with it no longer reach the HNQ and no longer draw as many bad answers.
Movies & TV
A new M. Night Shyamalan film, Foo in the Land of Bars, has come out. People would like to see it without having the twist and double-twist ending spoiled for them, but people who have already seen the film are asking questions left and right, not always being careful with the titles.
Mods on Movies.SE add foo-in-the-land-of-bars to the list of HNQ-banned tags. Questions tagged with it no longer reach the HNQ and no longer spoil other users.
A question on the new Quux Beta has reached the HNQ, but is actually a bad representation of Quux.SE. There's discussion in the Quux chatroom and on Quux.meta.SE if the question should be closed. Meanwhile, users across the network are getting their first impressions of the Quux Beta through that question.
Mods pro-tem on Quux.SE add exclude-from-hnq to the list of HNQ-banned tags and tag the contentious question with it. This questions is dropped from the HNQ and the new users of Quux.SE can discuss its topicality without fear of leaving the wrong impression across the network.