If your media system is connected to the TV, it depends on whether you are using X windows or a framebuffer to display graphics.
If you're using X windows, then since you DON'T want the output on YOUR machine, then you need to be logged in as the SAME user the X windows system is logged in as, (it's a permissions thing really).
Enter this environment variable export command in your ssh session to have X11 requests redirected to the local machine (the media box)
This will cause mplayer to be redirected to the local media-box video, you'll still be able to control mplayer with the ssh connection using the various keyboard commands it supports. (f for fullscreen (or -fs option), q-quit, p-pause, etc...)
If you're using X11 to display, check out the 'xhost +localhost' command and possibly re-enabling the insecure unrestricted port 6000 access for your Xserver. Again, depends on how you set up your media box.
if you're not using X windows, it'll probably work using the framebuffer output option of mplayer.
Try using mplayer with the '-vo fbdev' or '-vo fbdev2' option, telling mplayer to use the framebuffer to display output. Check the mplayer manual for more details. Those are the most likely to work, but there are other options, but they get more complicated.