Changing the favicon because of such a small issue sounds silly to me. If you need to be notified of new inbox messages, consider the next Stackapps:

- [stacknotify google chrome extension][1]
- [A user script to update the page/tab title with # of unread notifications][2]

If you want to, I can create a userscript that colors the whole bar red on new inbox messages.

 [1]: https://stackapps.com/q/2356/6969
 [2]: https://stackapps.com/q/2362/6969