I would like to see a badge (or, better, badges) for signing in to a Stack Exchange site with one of the native apps.
These would be bronze badges, and can only be awarded once per site. I'd expect they would be "Participation" badges.
- Green Robot
- Apple Fan
(These are probably the worst badge names ever, but we need to start somewhere.)
I would think that signing in to a site once with the native app would be enough to award the badge, but perhaps a criteria akin to "Enthusiast" could be used (i.e., "visit 30 consecutive days by app").