Making downvotes count for more
Downvotes could be weighted more heavily in the formula (at present one downvote is worth the same as one upvote). I would guess that this is unlikely to be done unless it is agreed to do it network-wide. This only helps with the problem of poor quality questions reaching HNQ. It does nothing for the many questions that reach HNQ without a single downvote, as they are good questions just not time consuming to answer. Whether such questions should be the main advert for our site is likely to divide opinion, and including downvotes in the discussion makes it harder to judge the split of that opinion. Hopefully answers here can address those two issues independently.
Excluding specific tags
It may be possible to stop certain tags from appearing in the HNQ list. There is already a list of words that are not permitted in titles in HNQ. Questions with a title containing such a word simply do not appear. However, the forbidden words are network-wide, whereas excluding a tag would be site specific. I'm not sure if this would be a problem, but if the tag we want to exclude never appears on other sites then it should be easier.
An example of a tag that makes a question likely to reach HNQ is answer-chaining. However, to me that tag seems like one which it makes sense to keep on HNQ since the high number of answers does not indicate low quality, nor attract users that would skew our voting and erode our scope definition.
Opting out of Hot Network Questions
Each site has a manually adjustable weighting that is currently used to adjust for the traffic level of a site (so that HNQ doesn't fill up with only Stack Overflow questions). Since this is site specific it might be easier to get this changed. It won't correct for any of the perceived biases but it could be used to simply opt out of appearing on HNQ as a site, if the community decided that was best. We'd still need to convince the Community Managers that this is also not detrimental to other sites in the network (there may be an argument that the high variety of sites in the HNQ side bar is healthy for SE as a network).
Is there really a problem?
Although it's good to ask the question, and good to see possible things that could change, I think it makes sense to gather meaningful data before making any decisions. I'd like to see suggestions of what statistics would best help decide whether there is a problem.
It's informative to look through the various discussions that have taken place on Mother Meta. There are many different opinions expressed and many of them sound plausible even though they contradict each other.
What I'd like to see is a HNQ history. The ability to call up statistics on questions based on the maximum hotness score they achieved, the length of time they spent in the HNQ list, and the voting data showing the proportions of votes that came from users at different rep levels (including votes from passing users who had never gained rep on the site beforehand). Even better would be the ability to see what proportion of users new to the site went on to gain rep there later.
I don't know how much of this could be made available without starting to unintentionally open routes to mining data on specific users, so this might need to be a mod only reporting tool (or even a community manager only tool) but I think its existence would be beneficial to everyone, regardless of who happens to be using it.
Easy and quick are not synonyms
It's possible to have a question requiring very little skill or innovation, but still taking a long time to write an answer for. Equally it's possible to have a question which attracts answers very quickly, but mainly from ingenious users who happen to be able to solve tough problems at high pace.
A question being in HNQ suggests that it is quick to answer. Whether it is also easy to answer will vary from question to question, and what the balance is would be interesting to try and measure. In the absence of definitive stats, I'd be interested to hear if anyone has examples from either side: HNQ questions that were easy and/or answerable by visitors, and HNQ questions that were hard and/or answerable mainly by existing users, or even only by a select few existing users.