What can I do with an Nexus One with a broken LCD? I've managed to acquire a Google Nexus One, that had an unfortunate encounter with a car door.  The screen still accepts touch input, but the LCD itself is broken - no display at all.  What is something "cool" I can do with it?  My main concern is, is it possible to root it and get a remote shell without being able to see the screen?
You can get a remote shell with it, without the screen working. You will have to install the android SDK on your PC. 

Another thing you can do is use Android ScreenCast to view the screen remotely. It doesn't have the best frame rate, but at least you will be able to interact with the screen. You may have to use this to get the remote shell working, as you may have to turn on USB Debugging in Settings -> Applications -> Development -> USB Debugging

You can also get a replacement screen, I found a couple on ebay, ranging from ~$50 - ~$100. Some are just the LCD screen, some include the the touch input panel too.