Multisite favicon.ico

I hope you are using different theme in each site

Just add this code in your header.php file.

<link rel="shortcut icon" href="https://wordpress.stackexchange.com/questions/25736/<?php bloginfo("stylesheet_directory'); ?>/favicon.ico" />

Add your favicon in each theme.

For favicon in admin area.

Add the following function in your function.php file.

function admin_favicon() { ?> 
<link rel="shortcut icon" href="https://wordpress.stackexchange.com/questions/25736/<?php bloginfo("stylesheet_directory'); ?>/favicon.ico" /> <?php } 
add_action('admin_head', 'admin_favicon');