This question is in some ways the inverse of Add a feature to migrate a comment into an answer .
A mistake I see many new users make on their questions is to respond to comments on their questions by posting an "answer" which should be either a follow-up comment or, if it's longer (e.g. adding sample code) as an edit to their question. They do this because they do not realize they can edit questions.
I participate on a different Q&A site which runs an open-source Q&A site framework, and it has (for users with enough rep) a "repost answer as comment" button:
For SE the above might not work as-is, since many (non-)answers could be too long to be a comment. There could be both options: for short non-answers they could be re-posted as a comment. For longer non-answers it could automatically append an edit to their question with some appropriate text indicating where the edit came from. Something like:
Updated (dd/mm/yyyy hh:mm via migration from answer):
[text of non-answer follows]
This can be done manually of course, but I see this problem often enough that it might be helpful to have an automated approach.
Or, if nothing else, perhaps a tool-tip/flash notice to new-users when they are about to post an answer to their own question that they can also edit their question to provide new information.