Create a copy of database and renaming it using mdf & ldf files

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