I cannot make my computer hibernate. I altered the settings so that pressing the power button sends the computer into hibernate mode but when I actually press the button, nothing happens. Also, is there any way in which I can add the hibernate option to the drop down menu that appears when I click on the power option?
Yep, the hibernate option in Power panel doesn't work indeed (checked that myself). As it was mentioned here, it's possible to hibernate your PC via
sudo systemctl hibernate command, but impossible to do that via the elementary's power control panel. Long story short, after some discussion the devs have decided to remove Hibernate option from elementary OS.
I do not know how to add it into the menu dropdown, but I have managed to achieve hibernation and set it on the Lid close. Same way if it works for you, you can add it on the powerbutton hit just by edit the appropriate line according to the last title in my answer there - https://elementaryos.stackexchange.com/a/21812/12483