Robin:

Interior wood stain is NOTHING MORE than dye dissolved in mineral spirits or alcohol. Any dried up wood stain on your hands will dissolve readily in mineral spirits or alcohol (depending on what the original stain carrier fluid was).

EXTERIOR wood stains will have some alkyd or acrylic binder in them so that they form a film over the wood. The purpose of having that binder film form over the wood is to encase the UV blockers and mildewcides which are also added to exterior stains to protect the wood from the Sun and mildew growth on it. Also, the binder film prevents the wood from absorbing liquid water at it's end grain, and that's what causes wood outdoors to split at it's end grain.

So, interior wood stain should come off easily with paint thinner and/or isopropyl or denatured alcohol. Exterior wood stain will come off with whatever cuts the alkyd or acrylic binder; and if the binder has only been dry for an hour or less, then water for water based exterior stains or mineral spirits for oil based exterior stains will get the stain off your hands.

While cooking oil might very well work, there are cheaper and more effective alternatives available to you.