Update: 28th Feb 2010 - Due to the reported problems people have been having with the existing approach below, I have created the following program for MozyHome & MozyPro. It runs in the system tray and checks various Mozy registry keys (not just the one below), and if they have been changed by Mozy, it resets them back to the values that allow Mozy to backup and restore from the network.
Download the 'Mozy Network Share Enabler' here
Download the 'Mozy Network Share Enabler' here (64 Bit only) version.
This is a free program, and is not supported by runPCrun or Mozy, and it's use is at your own risk. It currently works with MozyHome 1.16.4.0 and MozyPro 1.14.1.6.
If you'd like to thank me, there are a few options.

runPCrun recommend MozyPro for our business clients (usually in conjunction with an in-house backup solution, you can never have too many backups and they are faster to recover larger data volumes). However MozyHome* is a good product in itself for home users and I use it myself for my personal data.
When I initially tested MozyHome, there was a "add network share (BETA)" option which could be enabled. Since I use a Networked Attached Storage (NAS) device for my personal data at home (so my wife can easily access our photos etc.) this was perfect and so I subscribed. Unfortunately updates to MozyHome since have removed this option. When I contacted MozyHome support their (obvious) answer was that "we don't support that".
Since I was loathe to stop my subscription (it was only a month or so before that I'd finally uploaded 40Gb of my personal photos) I had a root around and found that while MozyHome had removed the option from the GUI, there was still an entry in the registry for this option.
All you have to do is open regedit (type 'regedit' at the Start > Run in XP or Windows button > Start search in Vista) and edit the following key
HKEY_LOCAL_MACHINE\SOFTWARE\mozy\optionsnetworksharesupportValue data REG_DWORD = 1
or with MozyPro Desktop Edition (the Server edition comes with network support)
HKEY_LOCAL_MACHINE\SOFTWARE\mozypro\optionsnetworksharesupportValue data REG_DWORD = 1
Now when you right click on the top level of the file system

When you click on this, the Add Network Share dialog box pops up.
And the share pops up now in your File System view for you to select what needs to be backed up.
Once backed up, it will also pop up in the Restore view.

Please note that I've found with the last few updates when MozyHome updates itself, it resets this value to 0, but the backup still functions properly. However it doesn't allow you to change the backup settings until the value is set back to 1 and the UNC share re-added. The restore view always works.
Hopefully MozyHome will not change this policy or remove it completely in future updates.
*This has some referral or partnership with this supplier via the link given. Check out our stance on reviews and recommendations.