Copy & rename db so I can attach both copies
If You want to use the same directory for your mdf\ldf files You will have to (physically) rename them.
1. Detach files from local PC
2. Rename mdf and ldf files
3. Copy\Move the files to the server directory where the original Database is stored.
(In My example I used D:\Data for mdf and E:\Logs for ldf, change the path before running the script)
4. Use this code:   

USE [master]
GO
CREATE DATABASE [myNewSite_db] ON 
( FILENAME = N'D:\Data\myNewSite_db.mdf' ),
( FILENAME = N'E:\Logs\myNewSite_db_log.ldf' )
FOR ATTACH
GO


5. If You want to change the logical file name to be the same as the physical one use:
(Assuming the original logical file name was "mysite_db")  

USE [myNewSite_db]
GO
ALTER DATABASE [myNewSite_db] 
    MODIFY FILE (NAME=N'mysite_db', NEWNAME=N'myNewSite_db')
GO
ALTER DATABASE [myNewSite_db] 
    MODIFY FILE (NAME=N'mysite_db_log', NEWNAME=N'myNewSite_db_log')
GO


Good luck,
Roi