How to rename a #variable in #drupal

yourmodule.module

<?php

/**
 * Renames a function
 */
function yourmodule_default_theme_settings() { 
  $new_variable_name = variable_get('old_variable_name', NULL); //Loads the variable to be renamed from drupal's variables table and assigns it to a new one.
  variable_set('new_variable_name', $new_variable_name); //Stores $new_variable_name into drupal's variables table.
  variable_del('old_variable_name'); //Deletes the old variable from drupal's variables table
}

It can be also usefull to call this function from yourmodule's hook_install

function yourmodule_install() {
  yourmodule_default_theme_settings();
}

and hook_update:

/**
 * Your message goes here.
 */
function yourmodule_update_7001() {
  yourmodule_default_theme_settings();
}