The Ubuntu 12.04 Precise Pangolin Operating System does not have any Hibernate button in the Shutdown menu. Hibernate is disabled by default because it does not work properly on some hardware.
Hibernate vs Suspend:
Both hibernate and suspend save the open session and the data and keeps your aplicationes opened, so that they can easily resumed. The difference is that the suspend option keeps the system turned on and drains a small amout of power, while hibernate saves everything to the hard disk and shuts down your computer.
To send your computer to hibernation, from the command line interface, use: sudo pm-hibernate.
If your computer restores your previous session, when you turn it back on from hibernation, that means the hibernation worked properly on your architecture.
If hibernation works on your system properly, you can add the hibernate button to your Ubuntu box.
Open the /etc/polkit-1/localauthority/50-local.d/com.ubuntu.enable-hibernate.pkla file in your favourite text editor and paste this: