At file useful-banner-manager.php, line #112:
add_menu_page( __( 'Banners', 'useful-banner-manager' ), __( 'Banner Manager', 'useful-banner-manager' ), 'manage_options', 'useful-banner-manager/useful-banner-manager-banners.php' );
plugin defines its own menu and uses 'manage_opitons' capability to restrict access to it. That is plugins author supposed that "Administrator" uses his plugins.
If you wish to change it you should edit this file. For example, create new role, add capabilities 'read' and new one 'useful-banners' to that role. Assign that role to the user. That user will see 2 menu items only: user profile and "Banners".
The problem is that if you wish that user has access to the plugin's widget you should give him 'edit_theme_options' capability, which is little wider than access to one plugin only.