I am actually looking for this same answer. I have learned that I can hibernate my non-root user logins.
However, I would like to be able to hibernate my root login as well. Any help?
In the task tray, or whatever it would be called in *nix, there is an image of either a power plug, or a battery. Right-click on this and you should have an option to Suspend to RAM, and also an option to Suspend to Disk. Suspend to Disk is equivalent to hibernate in Windows, and Suspend to RAM is equivalent to Standby in Windows.