The SE team has made several promising changes to the "Hot Network Questions" system. If you have previously found HNQs to be problematic, you are invited to participate in the discussion over there.
The purpose of this discussion is a new moderator superpower:
Moderators have the ability to remove questions from the HNQ List.
There are times when the hotness formula selects a question that a site would rather not have featured. Up until now, the only recourse that was available was to close the question (which may be appropriate anyway but isn't ideal when done purely to manage traffic), or to do nothing. We're putting the power in the hands of our moderators to remove questions that don't set a good example for their sites. I recommend each site have a meta discussion with guidance for moderators about when - if ever - a question should be removed.
I'd like to propose that we not invent a new policy on these questions de novo. I would prefer instead that we think as a community about the kinds of standards that we already have in place and communicate via voting, close-voting, and flagging, and how this new tool fits into our kit.
I'm imagining a process like this:
Over the next few weeks you will, like usual, notice some unusually active questions and think "well, that shouldn't be HNQ."
When you notice this, raise a custom moderator flag on the question and try to articulate what your reasoning is.
The moderation team will deal with your flag promptly --- that's the point of flags. But we'll also save your reasoning someplace, and as we start to get enough flags to divine a pattern, we'll try to summarize it either here or in a new post.
You could also link to posts in answers to this question, and explain how you think that the Hot Network Questions list has affected them. Those kinds of answers shouldn't be limited to bad posts that should be un-HNQ-ed: part of deciding what our policy should be will be deciding what we do want to keep. I'm especially interested in articulating, as a community, what types of posts get better responses after exposure via HNQ.
I also suggest flagging (or linking) current questions, because there will be some temptation to base any policy we develop on remembered examples from the past. For deciding how we think the site ought to work, it's usually better to notice how we are responding when we are using the site organically. I would hate for us to have a contentious discussion about a problematic question from the past and lock ourselves into some "policy" based on behaviors that don't really occur on the site any more. That's why I've titled this post "data collection": I'm proposing that we spend a reasonable amount of time just looking for patterns, and try to come to a conclusion around May-ish.
Update 2022: To see what our community keeps or removes from the HNQ in practice, consider this Data Explorer query.