I'm not sure about the timing on calling $collection->getSelect()->distinct(); - if your products aren't assigned to multiple categories that bit is unnecessary.

This isn't tested, so take it with a grain of salt, but I think it should work :

class Mage_Catalog_Block_Product_List_Random extends Mage_Catalog_Block_Product_List
{
    protected function _getProductCollection()
    {
        if (is_null($this->_productCollection)) {
            $categoryID = $this->getCategoryId();
            if($categoryID)
            {
                // Get array of subcategory ids
                $subCatArray = array();
                $catModel = Mage::getModel('catalog/category');
                $children = $catModel->getChildrenCategories($categoryID);
                foreach ($children as $subcat) {
                    $id = $subcat->getId();
                    array_push($subCatArray, $id);
                }
                // Use the array to filter the collection
                $collection = Mage::getModel('catalog/product')
                    ->getCollection()
                    ->joinField(
                        'category_id', 'catalog/category_product', 'category_id',
                        'product_id = entity_id', null, 'left'
                    )
                    ->addAttributeToSelect('*')
                    ->addAttributeToFilter('category_id', array(
                        array('in' => $subCatArray),
                    ));
                //Add Distinct to Select in case you have products in more than one of the child categories.
                $collection->getSelect()->distinct();
            } else {
                $collection = Mage::getResourceModel('catalog/product_collection');
            }
            Mage::getModel('catalog/layer')->prepareProductCollection($collection);
            $collection->getSelect()->order('rand()');
            $collection->addStoreFilter();
            $numProducts = $this->getNumProducts() ? $this->getNumProducts() : 3;
            $collection->setPage(1, $numProducts)->load();

            $this->_productCollection = $collection;
        }
        return $this->_productCollection;
    }
}

This isn't tested, so take it with a grain of salt, but I think it should work : EDIT : Code revised.

class Mage_Catalog_Block_Product_List_Random extends Mage_Catalog_Block_Product_List
{
    protected function _getProductCollection()
    {
        if (is_null($this->_productCollection)) {
            $categoryID = $this->getCategoryId();
            if($categoryID)
            {
                // Get array of subcategory ids
                $catModel = Mage::getModel('catalog/category');
                $parentCat = $catModel->load($categoryID);
                $children= explode(',',$parentCat->getChildren());
                 

                // Use the array to filter the collection
                $collection = Mage::getModel('catalog/product')->getCollection();
 
                $collection->joinField(
                    'category_id', 'catalog/category_product', 'category_id',
                    'product_id = entity_id', null, 'left'
                )
                    ->addAttributeToFilter('category_id', array('in' => $children))
                    ->addAttributeToSelect('*');
            $collection->getSelect()->group('entity_id');
            } else {
                $collection = Mage::getResourceModel('catalog/product_collection');
            }
            Mage::getModel('catalog/layer')->prepareProductCollection($collection);
            $collection->getSelect()->order('rand()');
            $collection->addStoreFilter();
            $numProducts = $this->getNumProducts() ? $this->getNumProducts() : 3;
            $collection->setPage(1, $numProducts)->load();

            $this->_productCollection = $collection;
        }
        return $this->_productCollection;
    }
}