Hibernation is a somewhat controversial subject in the Linux desktop community. It is considered to be difficult to implement and to have the potential to cause a lot of problems. I have used hibernation wherever possible on all my Linux desktops but it is not always straightforward. Older hardware may not support it, and even where the hardware is modern, changing a card could be enough to spike hibernation and require a reinstallation to fix.
Here are the instructions for changing the settings to enable hibernation on Xubuntu. These have been verified as applicable to 16.04 LTS.
Hibernation uses the swap partition to store the memory contents of the computer. You should ensure the swap partition is at least twice the size of the computer’s RAM.
My experience has been that older computers such as my dedicated work computers, which are both of advanced age, will not hibernate reliably in any OS. However, Xubuntu hibernates well on the four year old Ivy Bridge and two year old Haswell Refresh computers.