Change the app-name sent by docker's syslog driver
You can't do it. Here's a pending PR to add that feature: https://github.com/docker/docker/pull/12668
Hopefully it gets merged in soon. You could always roll your own, I suppose.
Update: Looks like this is slated for Docker 1.8
Update: This is now possible:
docker run --name my_container --log-driver=syslog --log-opt syslog-tag=my_application