Change WordPress plugin widget `id_base` without breaking widget

You can deprecate the whole function which will probably be quite hard to do in your situation but it is possible with some work:

You could check for both, but load the new one for future installs:

function badly_named() {

    __doing_it_wrong( 'badly_named', 'This method has been deprecated in favor of better_named_function' );

    /**
     * Call the better named method
     */
     better_named_function();
}