I had a situation where the Delete button was grayed out / disabled when attempting to delete a user account on a Windows 7 machine. This occurred when logged in with an account with full administrator privileges. The solution was to first navigate to the c:\Users\folder, find the desired user’s folder and try to enter it. Windows should then prompt you:
You don’t currently have permission to access this folder. Click Continue to permanently get access to this folder.
Once that is complete and you can access the folder, you should then be able to go back to Advanced System Settings and delete the user profile.
If you still have the issue, make sure your account has full administrative privileges (e.g., if you had to elevate your domain account) and then reboot the system.