Badge next to user's name

If I'm not mistaken it is just adding an image for a badge to a role.  You can do this under Administration > Membership > Roles > Edit a given role > click Upload File under the Role Avatar section.

Thanks, Matt Nevill. Unfortunately, this just adds it to the rest of the achievement badges: 

(side note, it also shows the role name when hovering, but our roles are passed through SSO, so it's not meant to be public-facing)

However, I'd like to have a specific badge show here to show that as the "important" one:

Hi Alex. My apologies, I completely missed you were talking about a smaller icon not reliant on the mouse hover. Hopefully someone else can answer if that's possible out of the box.

Alex Nassi   Social Edge Consulting edited the "Leaderboard" widget to also display the role badge. I can demo this for you if you are interested.

Thanks for attending Alex Nassi ! Looks like Matt Nevill and Gary Lungarini  have provided the correct answers. You need to follow Matt's steps and then modify the leaderboard to display the username.

Thanks lallison and Matt Nevill for your help on this! I wanted to follow up and confirm that  Gary Lungarini and I met about this earlier today to discuss what can be done. The team at Social Edge is scoping out the project.