Capability to edit post slugs

If it’s supposed to edit its own post only, make it with author‘s capabilities. If it should edit posts of other users, as well, make it with editor‘s capabilities.

Case 1

function wpse_add_your_role() {
    $role = get_role( 'author' );
    if ( ! empty( get_role( 'your_role' ) ) ) { 
        remove_role( 'your_role' );
    }
    add_role( 'your_role', 'Your Role', $role->capabilities );
}
add_action( 'init', 'wpse_add_your_role', 10 );

Case 2

function wpse_add_your_role() {
    $role = get_role( 'editor' );
    if ( ! empty( get_role( 'your_role' ) ) ) { 
        remove_role( 'your_role' );
    }
    add_role( 'your_role', 'Your Role', $role->capabilities );
}
add_action( 'init', 'wpse_add_your_role', 10 );

Code location

In both cases, put the code in current theme’s functions.php file. Once the new role is created, remove the code from functions.php, or comment out this line:

// add_action( 'init', 'wpse_add_your_role', 10 );

Final notes

You can fine tune your role, using $role->add_cap() or $role->remove_cap() methods, according to your needs. In case 1, do not remove edit_posts capability. In case 2, do not remove, neither, edit_posts nor edit_others_posts capabilities.