I'm trying to understand how the favorite button is meant to be used. I frequently see poorly written questions marked as "favorite". It seems the OP can make this designation. I was unable to find anything on this topic in a quick search of the online stackexchange documentation, and I'm not clear if the button plays any useful role. Apologies if I've missed something I should have read.
This feature request has been posted on main meta a few years ago: Add a page explaining the favorites feature to the Help Center. Existence of this feature request suggests that you are probably not the only person who searched for information on favorites unsuccessfully in the help center.
Very briefly: If you favorite a question (=if you click on the star next to it), then it is added to your favorites tab. The number near this star shows how many users favorited the particular question.
So basically all it does is that it helps you to create a list of questions which you want to have accessible somewhere and where you are able to follow relatively easily whether there has been any new activity (edits, new answers, new comments). So perhaps reasonable names for this feature could also be follow or bookmark, rather than favorite.
More often than not the users favorite question which are interesting to them. So typically the questions which are favorited by many users are good and interesting questions. It seems that you are asking mainly why some users favorite low quality questions. There might be many reasons for this. Some examples I can think of:
- I voted to close this question, since it was unclear/low quality. If the user edits the question, I want to see that there were changes, so that I can vote to reopen if it was sufficiently improved.
- I downvoted the question - if the OP edits the question, I might remove my downvote. I favorited it so that I notice any changes.
- The question is poor, but one the answers contains information which is interesting to me. I want to keep the link somewhere, so I add it to my favorites.
- The question is likely to be deleted, which makes finding it again difficult. Since I want to keep the link because of rather interesting exchange in comments, I favorite the question.1
- This question is rather easy, but it suggests a generalization which seems rather intriguing to me. At the moment I do not have too much time to think about this - I will favorite the question, so that when I check my favorites, I am reminded about this.
- This question is so poor that it definitely should be downvoted. I have already used all my 40 votes today, so I'll mark it and get back to it tomorrow.
I think that this post is worth editing. But since it is advised not to bump too many posts at the same time, I decide to do that later. So I want to mark the post somewhere. Favoriting the question is a possible a way to do this.
1Deleted questions are still shown on favorite tab - but if I remember correctly, they are displayed only to the users who are allowed to view them. Which means that the OP sees their own questions among favorites, even if they were deleted. And 10k+ users can see in this tab all deleted question they favorited. I have just favorited a few questions which seem likely to be deleted just to test this. (If you look in my favorite tab and you are 10k+ user, you should see some deleted questions there. As I am below 10k, I only see those which are not deleted.)
If you look at answers to the questions Why favorite but not upvote? and Would you ever favourite a question you down vote? on meta.math.SE, you will find some suggestions along similar lines. Various categories mentioned in this feature request on main meta can also give insight on how various users use favorites: Favorites improvements - search, categorize, personal tags, add note, favorite answer.
So that it is easier to examine low quality questions which are favorited (if somebody wants to), I have created these SEDE queries: Questions which are closed and favorited, Questions with negative score which are favorited Closed questions with negative score which are favorited. (And it should not be difficult to create other minor variations of these queries.)
In the cases where the question is favorited by single user, it might be the OP. Here is a SEDE query: Closed questions with negative score favorited only by the OP. If it is a relatively new and inexperienced user, it is quite plausible that sometimes they favorite their own question without really knowing what they are doing and how the favorites work. Another possible explanation is that if the OP sees that their question might be deleted, they can still keep the link to the question in this way.