One thing that stifles these kinds of requests is the fact that badges—in their current implementation—are rolled out to the entire network of Stack Exchange sites. So asking for Arqade-specific badges results in one of two scenarios:
- Badges rolled out to 80+ sites that don't make a lick of sense on those sites
- An overhaul of the badge system to support site-specific badges
Both are... less than ideal from a user and developer standpoint, respectively, so these types of requests are at best ignored and usually denied.
Rather than having contest-specific badges, what about having general contest participation badges? These would apply to any site that runs contests and promotions, which these days is a whole lot of them.
- Bronze badge: participated in one site-wide contest or promotion
- Silver badge: participated in (5|10) site-wide contests or promotions
- Gold badge: won a prize in a site-wide contest or promotion
What those badges should be called is left as an exercise for the reader, as I haven't a clue.