There are 2 major criteria used for badges and all badges satisfy at least one of these:
- Badges should encourage positive behaviors
- Badges should teach users about the features of the site.
I think this proposed badge would actually satisfy both criteria.
First it encourages positive behavior by having users help clean up their own posts. If a post isn't good, and a user recognizes it and they vote to close it they should be rewarded.
Second it teaches users about the features of the site. At 250 rep, users get the view close vote privilege, but with this privilege users also get the privilege to vote on their own posts (for closure) and I don't think many users are aware of this. A badge here would help users learn that this functionality is available. Which would help encourage the positive behavior.
As I mentioned in the comments, I think the post needs to be closed first before awarding the badge to prevent gaming, or there needs to be some other controls to help someone from voting to close for no reason just to get the badge.
As for a name, I don't know if
Honesty is an appropriate name as you aren't necessarily being honest for closing your own post.
Cleanup would work, but that is already a name for another badge, so maybe
Good Citizen since someone who closes their own post is being a good citizen.