It is called "spoofing" the drive. Basically you are taking a drive that is not meant to go into that Xbox and fooling the Xbox into thinking it should be there. Google "spoofing xbox 360 drive" and you will find tons of info.
I will give you this warning, I know someone who repairs 360's for a living and he has told me NEVER to do this. To go out and find the drive brand and model that it should have in it. Why? Because down the road it can cause the console to red ring. He said it is fairly common.