Aloha everyone,
I've never had any issues when renaming my computers before. Actually, you may have issues if you don't. By default, your HDD is called "Macintosh Disc" or something generic like that. Setting a different name allows for easier identification of specific computers. If you would like to share files over your home/work network, you can also go to
System Preferences -> Sharing
and click in the File Sharing checkbox to enable File Sharing. To better identify other computers/servers on your home/work network, you can use the text field in the top of that pane to change the name of your computer on the network. Then, when you press:
Shift-Cmd-K
you should see an icon for the other computers/servers on your home/work network, with the names specified in the Sharing pane.
With respect to the icon used, you'll need an .rsrc icon file for hard drives. Icons for hard drives are in a different format than those used for apps (.rsrc is for hard drives and .icns is for apps, if you want to change them once they've been installed). My former workflow was to use Icon Composer (installed if/when Xcode is installed - it's in /Developer/Applications/Utilities), then run the resulting .icns file through Icns2Rsrc.app. Unfortunately, Icns2Rsrc.app is a PPC program, and I have yet to find a suitable replacement. Fortunately for me, I have a late 2006 20" iMac that can not be upgraded to Lion (it only had a Core Duo CPU), so I can run Icns2Rsrc on that computer.
Here are the steps to change your hard drive icon. First right-click and copy the desired .rsrc icon file. Next, locate your hard drive in either the Finder or on your desktop, right-click and select Get Info. In the top-left of the Get Info window is the current icon being used for your hard drive. Now, click on that icon and you'll see a blue background around the icon. From the Finder menu in the menubar, select Edit -> Paste and you should see the icon change, both in the top left corner and in the Preview section of the Get Info window.
EDIT: It is possible to choose different names for your computer and primary hard drive, but I would advise against that, as it will probably cause too much confusion. Of course, if you develop a naming convention taking this into consideration, feel free to name differently.