WJWalton linked to this find: an official Dungeons & Dragons activity badge from the UK Boy Scouts program. At some point in time, it seems, there was a version of the Hobbies activity badge, intended as a catch-all for those pastimes Scouts already pursued that didn’t fall under the aegis of another activity badge. As commenter darrell explains below, this TSR-sponsored version of the Hobbies badge was awarded to all scouts, regardless of whether their hobby focused on Dungeons & Dragons.
So there’s some precedent for the Video Games badge Cub Scouts can earn. Really, there’s nothing untoward about either. Scouting’s always been about encouraging well-rounded development in all areas, outside and indoors. There are badges for studying architecture and nuclear science (!), as well as pioneering and personal fitness.
Between this and Walton’s take on a role-playing advocacy badge (scroll to the end of the article), I’m interested by the idea of having embroidered merit badge-like patches made up for general distribution.
For another perspective on role-playing and scouting, check out A Scoutmaster’s Blog, in which a Minnesota scoutmaster comments on his experiences role-playing, both as a player and a Scout and later a GM for his troop.