I think I must answer this question , piecing together a couple of things from my comments and other experiences that I've had on this site. I'll put it in a table format. I'll answer the more general question : what to do when you see a question with interesting mathematical content that could be closed under one of the MSE guidelines.
First, however, a caveat on what kind of questions you want to not be improving, or at least should not prioritize improvement of over doing other things. These are questions that will fall foul regardless of how you edit it. Namely :
You should get a feel for what is "inherently" off-topic content. In this case, you see a question but your doubt isn't about the context but about whether it's on topic or not (peer review of papers, open solution to conjecture, generalized advice etc. are the kind of questions that usually generate interest but also end up being closed as off-topic) then please write a meta post asking about it (if you're seeing something for the first time), and/or get a general feel for what's on-topic and what's not by reading meta. Alternately, answer/re-ask the question and be ready to defend it if and when it is closed as off-topic.
Broad questions : Don't edit questions that have multiple sub-questions within them even if there a lack of context for each question (like a series of PSQs) because additional context is not going to affect the broadness, and removal of content to reduce broadness prejudges the question that OP thinks is the most important or would benefit from the most, and hence isn't acceptable as an edit ( to me at least).
Duplicates : Please make sure before engaging with a question, that you check if it's a duplicate or not. Use Approach0, SearchOnMath, the standard meta duplicate pages or even Google the keywords to make sure that your question at least isn't a clear duplicate of another question. If you find a duplicate, then the low-quality question holds no worth and will probably be closed and deleted as such (rather than as a duplicate), although you can attach the link of the duplicate to the low-quality question perhaps. There's no point of context rewriting clear duplicates.
So if you find an interesting question that doesn't have any of the above issues but it only lacks context and/or standard details that you can supply in some form, then you are free to carry out some actions that will help preserve the content of the question. This I'll detail out in the table below, with (what I think are) pros and cons added on.
|Comment below the OP's question, asking for improvement. Pointers can be found at links here, here and here. Pointers may differ from source to background to hints for attempting the question.
||This is a gauge for OP involvement (if one wishes) to evaluate one's interest in reviving the question, gives OP a more hands-on role in improving the question, and they also get the benefit of an answer. Personally, a successful revival has an associated feel-good factor to it.
||Requires patience, may require constant dialogue with OP and rechecking of question for updates, and isn't useful if OP is suspended/removed/ not agreeing to add context (useless conversation). Can lead to frustration if e.g. someone answers the question while you are improving it, and it's not a very good answer.
|Insert context into the question and perform a rewrite : the rough guidelines are given here. Then wait for edit approval and (optional, but in your case I'd recommend it, Rounak) comment that you've added context ,with details of added context.
||The edit is in your hands, you get to instantly gratify yourself and (if the question is closed) send the question to the review queue for reopening so it can be addressed. The OP (if active) continues to be involved , and gets a demonstration of how their question can be improved.
||This is a gray area, and I myself am not very sure what goes through and what doesn't under this guideline : so debate will be there, and more than the previous option. You'll have to think harder about what to add and whether it changes OP's intent or not. Finally, this option is traditionally considered "rare" : one should formulate an argument that the question thus saved is of high quality, not necessarily via editing the question but during subsequent argument. In particular, it shouldn't be (rather obviously) closable for another reason (too broad/duplicate etc.)
|Write a new question : with your context, your motivation, and possibly your own attempt. Link to the old question, if the OP is active , maybe let them know you are asking this question. Credit images/links borrowed from OP's question if required. Merging questions, I spoke of in the comments but it's a rare event apparently, see here.
||The entire question is in your hands to shed new light on, to focus on different aspects of the question, and so on. You can dissociate from the issues in the original post. It's relatively risk-free and undebated as long as your question isn't in a position to be closed under a different close reason.
||I think you'll have to be more careful about the question being a duplicate or being too broad (because it's not worth the effort of a rewrite to see your question closed as well)than compared to the other options. The OP doesn't get the benefit of the answer if they're not around, and finally a bit of credit is taken away from them because they found the question in the first place.
All in all, you're thinking about many things when you see such a question , and you have to weigh them against one another :
You're weighing up how much the OP would benefit from seeing an improved version of their question rather than their activity and being hands-on with respect to question improvement.
You're weighing up your resources : being patient in a conversation with OP and waiting for their responses, versus the slightly quicker option of a context edit/creating a new question.
You're weighing up the kind of context that the question is missing : a context rewrite will restrict what content you can add to the question, while in your question you could write pretty much anything that motivated you.
Now, I've done all three , and these are my experiences :
Having an OP edit their question is immensely satisfying , particularly when you see that the same OP subsequently begins to post better questions so that you have someone you can depend on for great questions without having to worry about context. I've been involved in this process at least on 2-3 occasions, even if the questions were not particularly interesting to me. The only problem is when OP doesn't agree to edit the question : in that case, I would either leave the task of improvement to someone else or ask a new question with the same concept. I will also admit that it can emotionally drain me if I don't see something useful come out of this process or if someone else posts an answer and OP loses interest in improvement.
- I feel like I need to add this : "OP permission" is something that I like to do occasionally, but NOT overdo. It means to let OP know that you have useful information, to show it in a comment, and allow OP to confirm that this information is useful and can be added to the question. In that case, I wouldn't object against someone adding context, be it anything. However, this should NOT be overdone to the extent that the OP isn't in control of their own question : if one is taking advantage of the OP's desperation to not see the question closed by making these suggestions or wishes to retain content just to get reputation for answering it eventually, then it's bad practice. Most of the work SHOULD be done by the OP.
Performing a context rewrite, even on a fairly old question, can be controversial : this is probably because there's quite a bit of division about what is acceptable. An attempt is definitely not acceptable. The guidelines say that it's not right to modify the author's intent or misrepresent their mathematical maturity, and what goes beyond or stays before this line is anyone's call. Recently a question had a reference to the OEIS written into it (there was a sequence and the question was about how to calculate some terms), and enough people (including me) found that this was good enough to reopen the question.
A user found an (open) question that didn't have an answer and lacked context. They asked about it here. I gave them some resources, and a little background, and had them ask a new question. That was answered. Then the old was closed as a duplicate of the new. Everything that was done there should serve as an example if you want to be doing something similar. Note that the old question's OP wasn't alerted but they were relatively inactive : perhaps that is the only thing you need to do in addition to what the OP of that question did. I'd assume it was a very satisfactory experience for all involved and quite hassle-free. I'd say this is the best option!
A few other things :
You are free to comment on low-quality posts, registering your interest and encouraging OP and others to participate on the thread and ask for tips on improving it. Do not provide answers, but if the OP is interactive then it's not the worst thing in the world if an interactive discussion or a partial hint is able to resolve the question : even if it is low-quality. I don't know how the guidelines for context rewrites interact with the statement "comments from the OP can be lifted into the main post as context" but if this can be done (and I'll need others' opinions about it, to me it looks like a fairly unobjectionable thing to do) then consider reminding OP to lift their comments to the main post, or do it yourself with OP's permission.
If you think that a question does have sufficient context but get the feeling that you've seen such questions closed before, then you might want to ask a new question or answer the question and be ready to defend the question via a request in the meta reopen thread if and when it is closed, with an argument at hand. That's how the site should work. A community wiki answer is also an option, but it's used quite rarely and I can't pinpoint why it's used anywhere I've seen it, because in each such case the argument for not providing it was equally as good.
Low-rep users, particularly those who are interested in meta-moderation of the site and in expressing their site-related opinions, are somewhat hard done by the system and we can argue about this all day, but the point is that they can't cast close-reopen votes on questions, which are essential in deciding whether the content is MSE-appropriate or not. For these users, I would recommend asking a new question over editing context, for sure. Even in the event that there is some issue, the avenues for low-rep users to meta-participate and express opinions on questions range from creation of meta-posts about questions, to using the reopen-close request page, to commenting under the post announcing their support for certain questions, to flagging (this requires some minimal rep) if they think reviewers have not behaved appropriately with respect to a particular question.
So consult the pre-requisites, then consult the table, weigh up your priorities (things to add in question / OP-priority/ the time you have), and be ready to defend your choices. If you can do all these, then you are in a great position to revive interesting content, mould good question-askers, research content for questions, and make MSE a good place to be.
Note : If there is a very specific question or bunch of questions that you are looking at for editing currently, then attach them above and I can comment on what can be done/can't be done with these questions. These will of course, be my opinions.