When a Question gets closed and someone edits them to get them reopened, that should count towards a (bronze/silver/gold) badge.
I'm unsure if the OP of a question should count here as well, otherwise this could be gamed (write a bad question, wait it to be closed, edit it, have it reopened. Receive badge) otoh: After the process there is a good question.
To prevent people from waiting until its closed before they do their edit, this should maybe kick in after 3 close votes and a subsequent not-beeing-closed.
I think a bronze badge should be awarded for the first time ("First Aid"), a silver one ("Medic") at 100 and a gold one ("Saviour" or "MD") at 500 edits that led to reopening.
If multiple people edited a question before it gets reopened they all should receive this.
Just editing the tags should not count.
The Idea here is that people should be rewarded for not just "taking the trash out" by voting to close, but to try to aid the question if at all possible.
Edit and Reopenbutton?
Edit and Reopen" to fix grammatical mistakes to game the system for a badge.