Alpine Linux is the lightweight distribution built atop musl libc and BusyBox while using a GrSecurity-enhanced Linux kernel. Alpine Linux uses OpenRC as its init system. If you are unfamiliar with this "Small. Simple. Secure." distribution, you can learn more via AlpineLinux.org. The image for Alpine is a mere 5MB.
This blog post by Brian Christner points out that Docker creator Solomon Hykes recently confirmed they will be moving from Ubuntu to Alpine Linux as their operating system base. They have also hired the creator of Alpine Linux, Natanael Copa, as part of this effort.
More details as they become available.