I don't understand why you don't want to change it in Settings > General, but I'll show you how.
You're supposed to set your site title and description in Settings > General. Then, SEO and other plugins can use filters to modify those if needed. Here's a filter that changes the site title to "New Site Name"
function replace_name($name) {
$name = 'New Site Name';
return $name;
}
add_filter('option_blogname','replace_name');
Note that the old site title will still show up in Settings > General.
Another method would be to edit it directly in the database. In the table [table_prefix]options (by default wp_options), look for the option_name "blogname". The option value displays the site name. Changing this will update the site name in Settings > General.
One more time, the text boxes in Settings > General are designed so that you DON'T have to go through these methods.