How do I overwrite the text value?
You will need to implement mytheme_preprocess_rate_widget() and write your own version.

Notice how $variables['buttons'] is set halfway down the function in the original question. Since preprocess functions for a particular theme ALL get run, you can't (that I know of) stop the rate module running its preprocess function, but by re-assigning $variables['buttons'] in your own mytheme_preprocess_rate_widget() function, you will be able to control the output there. I would suggest something like:

function mymodule_preprocess_rate_widget(&$vars) {
  extract($vars);

  $buttons = array();
  foreach ($links as $link) {
    $button = theme('rate_button', 'my custom text here', $link['href'], 'rate-btn');
    $buttons[] = $button;
  }
  $vars['buttons'] = $buttons;
}