I have not been able to find a Firefox Extension to make Hotmail the default mailto client, here is a workaround:
1. Type about:config into your location bar and hit enter.
If you've never edited used about:config before, you'll see a warning.
2. Click "I'll be careful, I promise!" This will bring you to the about:config window.
3. In the filter field type "gecko". Double click the first entry gecko.handlerService.allowRegisterFromDifferentHost to change the value to true.
4. Click BOOKMARKS and BOOKMARK THIS PAGE, name it whatever you'd like.
5. Click BOOKMARKS, right click the bookmark you just created and click PROPERTIES
6. Copy and paste the text between the following text into the location box and click save: javascript:navigator.registerProtocolHandler('mailto','http://hotmail.msn.com/secure/start?action=compose&to=%s','Hotmail');
7. Click BOOKMARKS and open the bookmark you created
8. You will see an information bar drop down at the top of the window. Click "Add Application." You've finished installing the Hotmail protocol now you just need to select it.
9. Go to Firefox>Preferences>Applications, if you are using a Mac, or Tools>Options>Applications, if you are using Windows. Now scroll down the list by content type and find mailto. In the drop-down menu to the right select "Use Hotmail".
You should now be able to send mail using hotmail.
Source:
http://forums.mozillazine.org/viewtopic.php?f=7&t=2437529 and a few other sources