Some oils polymerise and set hard, others stay liquid indefinately, for treating wood idealy we want one that cures or sets. The oils that stay liquid can go rancid, olive oil often does this quite quickly. Of the oils that set they vary in how long they take but most natural oils take quite a while...several months. This is good if you are doing an oil painting where you want to keep reworking it over a period of months but maybe not so good if you want to start using your new kuksa. For this reason most commercial oils have drying agents added to make them set faster..the drying agents are mostly poisonous. The frustrating thing is that because these are not food products the manufacturers do not have to declare the ingredients so you don't know what you are getting. They also tend to add white spirit to thin the oil (and because they like to sell white spirit for £5 a litre) Often even on the supposedly food safe finishes there are also notes on what to do if you drink it or get it in your eyes...thats no good for me.
Of the natural oils walnut and linseed are the two that set fastest, I have heard that sunflower sets eventually but I have never known it to and have had bowls treated with it go rancid. Olive never sets.
Personally I would recommend walnut, I don't use it commercially due to the risk to nut allergy sufferers. A good alternative is linseed if you can source a pure pressed linseed, I get mine 5l at a time from horse feed merchants. Walnut is available in every supermarket in 250ml bottles alongside the olive oil. Stand the glass bottle in hot water for 5 mins before treating the wood, this thins the wood and helps it penetrate the wood, put plent on, let it soak a few minutes then wipe off the excess.
The other commonly used oils are mineral oils, also known as paraffin oil or liquid paraffin (the liquid version of paraffin wax candles) sold in many woodworking shops and IKEA as a finishing oil or your local chemist as a laxative.
This never sets so does not protect the wood or build a patina in the same way but it never goes rancid either. It doesn't seem right to me putting mineral oil on wood but it will do no harm.
Note the oils mentioned will take a few months to set properly and give a waterproof finish but that does not stop you using an item straight away, just keep it off you best linen tablecloth.