How to make a search tab the default tab ?

I would check out hook_menu_alter first.

Here is what I would try - I have done something similar with success recently:

function marc_union_menu_alter(&$menu) {
  $menu['search/node']['type'] = MENU_LOCAL_TASK;
  $menu['search/google_cse']['type'] = MENU_DEFAULT_LOCAL_TASK;
}

Check the exact menu path of the goodle_cse search.

I know this is an old thread, but I hope it will usefull for someone. Add next functions in your module file (the example is shown for modul's name 'search_files' and a path to search 'google_cse', replace it for your case):

/**
* Implementation of hook_form_FORM_ID_alter().
*/
function search_files_form_search_block_form_alter(&$form, &$form_state) {
    $form['#submit'][] = 'search_files_box_form_submit'; // rewrite drupal's function search_box_form_submit($form, &$form_state) 
}

/**
 * Process a block search form submission.
 */
function search_files_box_form_submit($form, &$form_state) {
  if (isset($_REQUEST['destination'])) {
    unset($_REQUEST['destination']);
  }
  if (isset($_REQUEST['edit']['destination'])) {
    unset($_REQUEST['edit']['destination']);
  }

  $form_id = $form['form_id']['#value'];
  $form_state['redirect'] = 'search/google_cse/'. trim($form_state['values'][$form_id]); // HERE WE MAKE A RIGHT REDIRECT
}

Here is a custom module I wrote to achieve the desired result you are looking for.

This module adds a setting in your administration theme to check a box to enable this functionality. Leave this option unchecked to use the default "Content" tab.

See the contents of my google_cse_default.module file below:

/**
* Module admin configuration settings
*/

function google_cse_default_form() {

  $form = array();
  
  $form['google_cse_default'] = array(
    '#type' => 'checkbox',
    '#title' => t('Show Google CSE as the default tab for site searches?'),
	'#default_value' => variable_get('google_cse_default', 0),
	'#description' => t('Check this box and save the configuration to show the Google CSE as the default tab for site searches.'),
  );

  return system_settings_form($form);
}

/**
* Module menu setup
*/

function google_cse_default_menu() {

  $items = array();

  $items['admin/settings/google_cse_default'] = array(
    'title' => 'Google CSE Default',
    'description' => 'This module will show Google CSE as the default tab for site searches.',
    'page callback' => 'drupal_get_form',
    'page arguments' => array('google_cse_default_form'),
    'access arguments' => array('Administer google cse default settings'),
    'type' => MENU_NORMAL_ITEM,
   );

  return $items;
}

/**
* Display help and module information
* @param path which path of the site we're displaying help
* @param arg array that holds the current path as would be returned from arg() function
* @return help text for the path
*/

function google_cse_default_help($path, $arg) {
  $output = '';  //declare your output variable
  switch ($path) {
    case "admin/help#google_cse_default":
      $output = '<p>'.  t("This module will show Google CSE as the default tab for site searches.") .'</p>';
      break;
  }
  return $output;
} // function google_cse_default_help

/**
* Valid permissions for this module
* @return array
*/
function google_cse_default_perm() {
  return array('Administer google cse default settings');
} // function google_cse_default_perm()

/**
 * Implementation of hook_form_alter()
 *
 *
 */
function google_cse_default_form_alter(&$form, $form_state, $form_id) {

  if ($form_id == 'search_form' || $form_id == 'search_theme_form') {
  
	if (variable_get('google_cse_default', 0)) { 
	
		$form['#submit'][] = 'google_cse_default_submit'; // rewrite drupal's function search_box_submit($form, &$form_state)
		
	}
  }
  
}

/**
 * Process the Google CSE serach redirect
 *
 *
 */
function google_cse_default_submit($form, &$form_state) {
	
	if (variable_get('google_cse_default', 0)) { 
	
	  if (isset($_REQUEST['destination'])) {
		unset($_REQUEST['destination']);
	  }
	  
	  if (isset($_REQUEST['edit']['destination'])) {
		unset($_REQUEST['edit']['destination']);
	  }

	  $form_id = $form['form_id']['#value'];
	  $form_state['redirect'] = 'search/google/'. trim($form_state['values'][$form_id]); // HERE WE MAKE A RIGHT REDIRECT
  
	}
}

In my google_cse_default.info file I set the module to depend upon the google_cse module.

See the contents of my google_cse_default.info file below:

name = Google CSE Default
description = "This module will show Google CSE as the default tab for site searches."
dependencies[] = search
dependencies[] = google_cse
core = 6.x