I apologize if this has already been suggested elsewhere; I did a cursory search that did not return any results. At the least, it looks like this suggestion does not have it's own question on Meta.
There has been quite a bit of buzz regarding the Hot Network Questions (HNQs), and what should and should not be included.
I would suggest that, regardless of what is decided, the HNQs should be customizable. Of course, there should be a default list / behavior of what is included in the HNQs, which would be used from anonymous users and by default.
This default list may exclude certain sites, for various reasons. This question isn't meant to discuss the merits of this, but to point out that the pain of these decisions on certain users (some users will be upset if a site is excluded, others will be upset if a site is included) may be alleviated by allowing users to customize which sites show up in the HNQs.
For example, (I believe) many of the language sites are excluded from the HNQs by default, as the questions are not relevant to most users. Well, if I speak Chinese, it would be great to include that site in my HNQs. Or, if I am only interested in StackExchange for programming / technical questions, I may wish to limit my list to only those sites.
I know this feature would come at a "cost", of reducing StackExchange's ability to advertise certain sites to certain users (the whole point of the HNQs!). However, if a user deselects a site from their HNQs, it's likely they aren't interested in the site and wouldn't visit it anyways. I would also argue they will now be more likely to pay attention to the HNQs and visit other sites that they are interested in.
EDIT: Regarding the marked duplicate, Filter “Hot Network Questions” by excluding sites... That question is similar, although it sparked answers closer to "here's a workaround you can implement to achieve this functionality", rather than a discussion of whether this functionality should be implemented into the site. I think it's also worth revisiting this idea, as it seems the answer 4-5 years ago was "We're going to try some other things to improve HNQs before considering this," and obviously those things have been tried, to varying levels of success. I also think that the recent events around HNQs give this idea more merit.
In addition, most of the previous suggestions came from a place of "I want to exclude a site from my HNQs", whereas, this one comes from both directions, including "I want to include an excluded site in my HNQs". For example, I really enjoy Interpersonal SE, and it, along with the Workplace SE, are my big reasons for looking at the HNQs. Since it has been excluded, I have (1) not spent as much time looking at the HNQs, and (2) almost stopped looking at Interpersonal SE, due to the inconvenience.
The idea of excluding sites or questions for non-practical reasons (i.e. a title not being in English) is a fairly new one, as far as I am aware, and I think it warrants a revisiting of this suggestion.