I would like to update my brief a little bit based on the submissions. I like the idea of using video or play themes (cameras, buttons, etc), but also don't want to be too literal. I'd prefer not to use cameras, and the use of lenses should be subtle. Play buttons are ok, but ideally they will also be subtle and stylized.
Another option may be to use the idea of looping. The app will allow clients to use content for second, third, and fourth (etc) loop uses. Maybe try using some sort of loop, looping, or spiral into the design? Again, not too literally, but as a part of the theme.
Also, I like the idea of having both an icon and the name in the logo. They don't have to bleed together, but certainly can.