I think "too broad" covers it.
The close banner on questions closed as "too broad" says:
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question.
Apart from the second sentence, this is actually pretty on-point for insufficiently specified story-ID questions. The way to save these questions by editing is indeed to limit them to be more specific and have enough detail to identify an adequate answer. Even the generic "How to Ask" page which the third sentence links to has some useful information:
Have you thoroughly searched for an answer before asking your question? Sharing your research helps everyone. Tell us what you found and why it didn’t meet your needs. This demonstrates that you’ve taken the time to try to help yourself, it saves us from reiterating obvious answers, and above all, it helps you get a more specific and relevant answer! [...]
If you ask a vague question, you’ll get a vague answer. But if you give us details and context, we can provide a useful answer.
The only extra advice we'd include in a custom close reason would be for the OP to read the information here or here and use the checklists to help jog their memory. This would be useful extra info, admittedly, but it's easy to provide that in a comment (which people may be more likely to read than a close banner, since it actually sends them a notification), and the "too broad" text is still good enough - maybe not the best we can do, but not totally irrelevant either. Which brings me to ...
Is this really worth a custom close reason?
A given SE site can have up to three custom close reasons (or up to five if the mods beg an employee to increase the threshold). We've already implemented one for recommendation questions, which has been used quite a lot since then - easily enough to justify its creation, IMO. Do we really want to use up another of those precious few slots for overly-vague ID questions?
So far at least, it hasn't been a big problem. We've had well over 100 ID questions, and only 5 of them have been closed for not being specific enough. (Links only visible to 2k+ rep users: one, two, three, four, five. Thanks to @Mithrandir for help finding all of these.) This workload seems easily small enough to handle by "too broad" votes and helpful comments. (You might argue that the workload will increase as the site grows. Anecdotally, I don't think it will - based on my experience at SFF, with 10,000 story-ID questions, where we've always used "too broad" and nobody's even proposed a custom close reason for vague ID questions.)
Also, several of us are already leaving comments on many ID questions - even those that don't get closed - linking the OP to the tag wiki info and asking them if they can remember any more details. We don't even need the same links in the close reason if they're already there in comments!