How to change Firefox icon shown on panel and menus?

To answer my own question:

Ignacio's answer takes care of the large icon on the system menu, and the taskbar launcher if you add one. Changing the icon on the taskbar button/panel button took more digging.

Uninstalling the 'native' Firefox from LMDE (sudo apt-get remove firefox) and installing the tar.gz from http://getfirefox.com still left me with the orange eyeball on the panel button.

So I started hunting through folders, and eventually found it: The icon I can't stand is in /usr/share/icons/Mint-X/apps/16/firefox-png.

I renamed the file to firefox-ugly-icon.png, and copied in the icon I want (from another system). Problem solved.