Here is a workaround that will get you two out of three. My favicon shows up in the browser tab and on the favorites bar but not in the address bar. See it here.
1. Design your icon and download it. I used http://www.favicon.cc/
2. Insert the image on one of your pages, just like adding any other image on Website Builder. Or upload to a host site like picasa or photobucket. I happened to have a photo credits page so that's where I put mine. It's so tiny, you could stick it in a bottom corner of any page on your site and it would probably go unnoticed.
3. Save and publish (if you added it to your web site).
4. Open up notepad or wordpad and paste this code there (we'll come back to it in a few seconds):
(Thanks to Michael's comment below,
we now have a much simpler code to work with.
Thank you Michael!)
6. Right click on the favicon image, click "Copy image URL"
7. Go back to the code you pasted in notepad. In the line that says href="http://example.com/myicon.png" , highlight the text between the quotations marks, right click and click paste. A long line of url text will replace the previous url. Make sure you still have the quotation marks.
8. Now copy your entire new code.
9. Log back into website builder.
10. Click the "manage site" button (just to the right of the green Publish button), choose "site settings", click the "general settings" tab.
11. Paste the code into an empty line in the black box under the HEAD section.
12. Click OK, save, publish.
CHROME: When you're finished, visit your page and hit F5 to refresh. Your browser is storing your previous version and will not show you your updated version until you refresh. New visitors to your page will see the new favicon but you will not unless you hit F5 to refresh. (Did I say refresh enough? Oh, there it was one more time). I use Chrome, by the way; this works.
NOTE: You can also use a 16x16 gif image. I did this because I wanted to edit my downloaded icon design a little further in GIMP (similar to photoshop but free software). Make sure your transparent parts stay transparent by pasting into a new layer and disabling the background layer before you save. You'll alter the code above by telling it you're using a gif file instead of an ico file: change... type="image/ico" ...to... type="image/gif".
Oh wow! I never noticed until now that we can change our blogger favicon from the orange and white B. It's a widget on the Layout page in the upper left corner titled (drum roll please)... Favicon.