Update Specific Key Value in Complex `wp_options` object

Sorry for being dense, you need to grab the object, overwrite the key you want to set and then save the updated object as the new option:

$my_plugin_settings = get_option('my_plugin_settings');
$my_plugin_settings->object_key = 'new_value';
update_option('my_plugin_settings', $my_plugin_settings);