To be clear, you are not looking to setup a Linux box to stream video from, you just want to be able to watch streaming video on it, right? And the only thing you want is a browser, no desktop environment?
You could set this up on most any desktop distribution. By copying a browser's .desktop file found in /usr/share/applications to /usr/share/applications/ you can log straight in to a browser.
You could also look into getting Chromium OS running on your machine, which would perform a similar function, probably better. »
chromeos.hexxeh.net/