Remove the "eDrive" on my desktop permanently.

An eDrive is usually created by TechTool Pro. See http://www.versiontracker.com/dyn/moreinfo/macosx/2895 Do you have TechTool Pro?

I would think you could drag the icon to the Trash & empty.

 Cheers, Tom 😉

This was installed by Micromat's TechTool Pro. In order to remove it you need to start up from the TechTool Pro DVD/CD and in the 'Tools' section select 'eDrive'; click the 'Remove eDrive' button.

If you boot from the Tech Tool Pro DVD/CD and click 'Remove eDrive,' Tech Tool Pro will UNINSTALL the eDrive. The eDrive is one of the great things about Tech Tool Pro. It allows you to boot your computer if your regular startup disk won't boot properly.

The eDrive icon should show up on your desktop only initially during startup. After the Finder completely loads, the eDrive icon should disappear. If it remains, I would suggest contacting Micromat support. The fact that the icon remains visible on the desktop is something they should know about, as it is not normal behavior. But they're the folks you need to check in with.

http://www.micromat.com/index.php?option=com_wrapper&Itemid=96

This did not work for me: I booted from my TechTool Pro version 4.6.1 DVD and tried to select "eDrive" in the "Tools" section. Nothing happens when I click on "eDrive".