Exclude main blog from get_blogs_of_user

If you drop this line in your code, you will see all the properties of $user_blogs.
echo '<pre>'.print_r($user_blogs,true).'</pre>';

One of them is userblog_id, so you just have to check against it before echoing the blogname.

<?php 
$user_blogs = get_blogs_of_user( $user_id ); 
if (!$user_blogs) {
    echo 'no blogs';
} else {
    echo '<div><ul>'; 
    foreach ( $user_blogs as $user_blog ) {
        if ( $user_blog->userblog_id != get_current_blog_id() ) { 
            echo '<li>' . $user_blog->blogname . '</li>';
        }
    }
    echo '</ul></div>'; 
}