Magento – Change magento backup folder from /var/backup to different/external folder

Maybe the var folder does not have the proper rights?

When the database grows bigger, the Magento backup function simply fails to do its job because it's too big to handle. Then you have to backup mysql directly.

the folder has always been var/backups.
See the method Mage_Backup_Helper_Data::getBackupsDir()

public function getBackupsDir()
{
    return Mage::getBaseDir('var') . DS . 'backups';
}

it's there in 1.9 and it has been there since the backup module was introduced.

Make sure the folder is writeable and you have enough disk space.

[EDIT]
I was half right in my original answer.
I assumed that Mage::getBaseDir('var') always returns MAGENTO_ROOT/var but that's not always true.
Take a look at the method: Mage_Core_Model_Config_Options::getVarDir() (this is called when determining the var dir).

public function getVarDir()
{
    //$dir = $this->getDataSetDefault('var_dir', $this->getBaseDir().DS.'var');
    $dir = isset($this->_data['var_dir']) ? $this->_data['var_dir']
        : $this->_data['base_dir'] . DS . self::VAR_DIRECTORY;
    if (!$this->createDirIfNotExists($dir)) {
        $dir = $this->getSysTmpDir().DS.'magento'.DS.'var';
        if (!$this->createDirIfNotExists($dir)) {
            throw new Mage_Core_Exception('Unable to find writable var_dir');
        }
    }
    return $dir;
}

If the var folder cannot be created magento uses as var dir the system tmp folder and creates magento/var inside it.
So for linux that should be /tmp/magento/var.
Take a look in there, maybe you will find your missing backups.