There are many pages out there explaining how to override Joomla Layout, or even full classes, but it's also possible to overrides functions called by JHtml::_() function.

To do that, all you need to do, is to place the function to override in a system plugin, and to register the function override in a onAfterInitialise() function.

here is an example (that forces the date picker widget to use Monday as the first day of the week):

<?php
/**
 * @package     Mypackage
 * @subpackage  Plugins
 *
 * @license     GNU General Public License version 2 or later, see LICENSE.
 */

// No direct access
defined('_JEXEC') or die('Restricted access');

// Import library dependencies
jimport('joomla.plugin.plugin');

use Joomla\Utilities\ArrayHelper;

/**
 * Jhtml override
 *
 * @since  1.0.0
 */
class PlgSystemJhtmloverride extends JPlugin
{
   public function onAfterInitialise()
   {
      JHtml::register('calendar', array('PlgSystemJhtmloverride', 'calendar'));
   }

   /**
    * Displays a calendar control field
    *
    * @param   string  $value    The date value
    * @param   string  $name     The name of the text field
    * @param   string  $id       The id of the text field
    * @param   string  $format   The date format
    * @param   mixed   $attribs  Additional HTML attributes
    *
    * @return  string  HTML markup for a calendar field
    *
    * @since   1.5
    */
   public static function calendar($value, $name, $id, $format = '%Y-%m-%d', $attribs = null)
   {
      static $done;

      if ($done === null)
      {
         $done = array();
      }

      $readonly = isset($attribs['readonly']) && $attribs['readonly'] == 'readonly';
      $disabled = isset($attribs['disabled']) && $attribs['disabled'] == 'disabled';

      if (is_array($attribs))
      {
         $attribs['class'] = isset($attribs['class']) ? $attribs['class'] : 'input-medium';
         $attribs['class'] = trim($attribs['class'] . ' hasTooltip');

         $attribs = ArrayHelper::toString($attribs);
      }

      JHtml::_('bootstrap.tooltip');

      // Format value when not nulldate ('0000-00-00 00:00:00'), otherwise blank it as it would result in 1970-01-01.
      if ($value && $value != JFactory::getDbo()->getNullDate() && strtotime($value) !== false)
      {
         $tz = date_default_timezone_get();
         date_default_timezone_set('UTC');
         $inputvalue = strftime($format, strtotime($value));
         date_default_timezone_set($tz);
      }
      else
      {
         $inputvalue = '';
      }

      // Load the calendar behavior
      JHtml::_('behavior.calendar');

      // Only display the triggers once for each control.
      if (!in_array($id, $done))
      {
         $document = JFactory::getDocument();
         $document
            ->addScriptDeclaration(
               'jQuery(document).ready(function($) {Calendar.setup({
         // Id of the input field
         inputField: "' . $id . '",
         // Format of the input field
         ifFormat: "' . $format . '",
         // Trigger for the calendar (button ID)
         button: "' . $id . '_img",
         // Alignment (defaults to "Bl")
         align: "Tl",
         singleClick: true,
         firstDay: 1
         });});'
            );
         $done[] = $id;
      }

      // Hide button using inline styles for readonly/disabled fields
      $btn_style = ($readonly || $disabled) ? ' style="display:none;"' : '';
      $div_class = (!$readonly && !$disabled) ? ' class="input-append"' : '';

      return '<div' . $div_class . '>TEST'
         . '<input type="text" title="' . ($inputvalue ? JHtml::_('date', $value, null, null) : '')
         . '" name="' . $name . '" id="' . $id . '" value="' . htmlspecialchars($inputvalue, ENT_COMPAT, 'UTF-8') . '" ' . $attribs . ' />'
         . '<button type="button" class="btn" id="' . $id . '_img"' . $btn_style . '><span class="icon-calendar"></span></button>'
         . '</div>';
   }
}