I'm not sure how you're doing it but it works that way from phpMyAdmin.
A typical sql dump from phpMyAdmin using my above settings:
-- phpMyAdmin SQL Dump
-- version 2.11.9.1
-- http://www.phpmyadmin.net
--
-- Host: localhost
-- Generation Time: Oct 07, 2008 at 10:20 AM
-- Server version: 4.1.22
-- PHP Version: 5.2.6
SET SQL_MODE="NO_AUTO_VALUE_ON_ZERO";
/*!40101 SET @OLD_CHARACTER_SET_CLIENT=@@CHARACTER_SET_CLIENT */;
/*!40101 SET @OLD_CHARACTER_SET_RESULTS=@@CHARACTER_SET_RESULTS */;
/*!40101 SET @OLD_COLLATION_CONNECTION=@@COLLATION_CONNECTION */;
/*!40101 SET NAMES utf8 */;
--
-- Database: `rumbaar_smf3`
--
-- --------------------------------------------------------
--
-- Table structure for table `smf_attachments`
--
DROP TABLE IF EXISTS `smf_attachments`;
CREATE TABLE IF NOT EXISTS `smf_attachments` (
`ID_ATTACH` int(10) unsigned NOT NULL auto_increment,
`ID_THUMB` int(10) unsigned NOT NULL default '0',
`ID_MSG` int(10) unsigned NOT NULL default '0',
`ID_MEMBER` mediumint(8) unsigned NOT NULL default '0',
`attachmentType` tinyint(3) unsigned NOT NULL default '0',
`filename` tinytext NOT NULL,
`size` int(10) unsigned NOT NULL default '0',
`downloads` mediumint(8) unsigned NOT NULL default '0',
`width` mediumint(8) unsigned NOT NULL default '0',
`height` mediumint(8) unsigned NOT NULL default '0',
PRIMARY KEY (`ID_ATTACH`),
UNIQUE KEY `ID_MEMBER` (`ID_MEMBER`,`ID_ATTACH`),
KEY `ID_MSG` (`ID_MSG`)
) ENGINE=MyISAM DEFAULT CHARSET=latin1 AUTO_INCREMENT=8 ;
The database name is only in comments and doesn't affect the target db name or importation procedure. You need drop tables as if you are importing into an already populated db or you'll get duplicate key issues. You need to turn off extended inserts as it will not work with BigDump.
I've performed dumps and re-importation without issue into differently named databases using this method on a number of occasions. Are you sure you're defining the correct attributes in the BigDump.php settings?
Also you should be able to call your db anything you want, though it might be prefixed with your account name, etc.