<p>Suppose someone worked for a company that put up an HTTP proxy preventing internet access without password authentication (NTLM, I think). Also suppose that this password rotated on a daily basis, which added very little security, but mostly served to annoy the employees. How would one get started writing a Firefox add-on that automatically entered these rotating passwords?</p>

<p><em>To clarify: This add-on would not just submit the password; the add-on would programmatically generate it with some knowledge of the password rotation scheme.</em></p>

This is built into Firefox. Open up about:config, search for 'ntlm'
The setting you're looking for is called network.automatic-ntlm-auth.trusted-uris and accepts a comma-space delimited list of your proxy server uris.
This will make FireFox automatically send hashed copies of your windows password to the proxy, which is disabled by default for obvious reasons. IE can do this automatically because it can use security zones to figure out whether a proxy server is trusted or not.
Blog post discussing this