This is what i did for mine just this second:
1.) Upload the .ico file it to this folder in your cPanel/website files:
public_html/websitename.com/wordpress/wp-content/themes/themename/images
(obviously choosing the ‘website name’ and ‘theme name’ relevant to the website and theme that your currently using and wish to add a favicon to. Make sure your website is currently using this theme as well…)
2.) Make a note of the favicon file name ie. MyLovelyFavicon3.ico
3.) Then return to your dashboard and click on ‘Appearance’ and then ‘Editor’.
4.) On the far right side there should be a list of .php files and a drop down menu at the top which lists the templates you have installed for WordPress. Ensure the template selected in this menu is the same theme as where you uploaded your favicon.ico file in your cPanel.
5.)Then looking at the list of php. files along the side as mentioned previously one should be called ‘header.php’. Click on this and a different page of code should open in the text box for you to view and edit.
6.) Then cut and paste the following code into it at the very bottom after all the other code…:
<link rel="shortcut icon" href="<?php bloginfo('template_directory'); ?>/images/favicon.ico" />
…changing the ‘favicon.ico‘ at the end to show the filename of your own .ico file i.e.
<link rel=”shortcut icon” href=”<?php bloginfo(‘template_directory’); ?>/images/MyLovelyFavicon3.ico” />
For me, I found that installing this code anywhere else within the exisiting code didnt work. I had to paste it at the very bottom apart from all the other code and then it did.
7.) Then click update and then view your website/refresh. The favicon should appear immediately.