I use camelia oil at the moment for razors that are in regular usage. I don't really know why, other than the fact that it was recommended for straight razors, and because the small bottle I bought will last the rest of my life! I find it to be no better than mineral oil, but as Bart said, all we need to do is keep moisture and air away from the steel.
For long term storage, I favor gun oil or a gun grease. The products designed for firearms have no additives that could possibly stain the metal, and additives are the things in oil we need to be wary of.
Years ago, the Trapshooters all started using Wolf's Head motor oil on their fine target guns. One of them had read in one of the magazines that the lubricity of this particular oil, never a top of the line product in it's niche to begin with, was better that the engineered oils designed for firearms.
Well, many had the highly polished actions of their guns stained by one of the additives in the oil. Not surprising when one thinks about it.. The internal moving parts of an automobile engine do not need to impress anyone with their highly polished beauty.
At any rate, I tend to stay with purpose designed products, or ones with the same objectives.