A Busybox docker image is useful if one is building a container for which busybox can fulfill its dependency chain without needing a full Linux distro.
Often, an embedded appliance can consist of nothing but a statically-linked copy of busybox, an init script that mounts procfs, sysfs, &c. with busybox-provided tools, and then the actual application being invoked. With docker setting up the filesystem namespace, even that init script isn’t necessarily needed.