I am wondering if it is possible to do a GPO Drive map based on the computer name, rather than the user name. I would like to do this to set up a backup share for each computer where I can place the backup files.

ex. COMPUTER1 would be mapped to \\fileserver\backups\COMPUTER1

I don't see why not. COMPUTERNAME is a system variable. I'm not 100% about mapping drives with the variable, but you can try it anyway.

It would be something like \\fileserver\backups\%COMPUTERNAME%

So simply, it would just be a GPO on Domain Computers that says to map drive X to \\servername\sharename\%COMPUTERNAME%?

I have a hunch that you may have to make that a batch file and use a GPO to run the file at logon. I've done similiar stuff with user home directories in VBS so it can't be much different.

Well, it was easy enough. No batch file needed. was just able to map the drive to \\SERVER\Backups\%COMPUTERNAME% within the GPO Manager in 2k8 R2.

Gotta love environment variables :)