A lot of specialty-labeled oils like reel oil, clipper oil, cutting board oil, sharpening stone oil, shredder oil, etc. are nothing more than what is usually a 'light' or 'white' grade of mineral oil, which usually is more refined and thinner viscosity (although the laxative-grade stuff is heavier, but still food-safe). A lot of the other 'light' or 'white' grades are also food-safe as well. They're very clean oils, both colorless and odorless. Look for 'USP' labeling on the product to specifically denote it's safe for food use. Any of them would be just fine for simple use as a knife pivot oil, and for keeping some rust protection on non-stainless blades.
I've liked Norton's Sharpening Stone Oil (edited to add: it's 100% light mineral oil and food-safe) both for use on my oil stones and at the same time, giving the knife's joints a little lubrication after sharpening. And I've also used a very similarly light & clean mineral oil I picked up at a local restaurant supply store. It's labeled for use with food processing equipment like slicers, blenders, mixers, etc. I've been using that on my stones and occasionally for lube purposes as well. None of these are particularly expensive for the volume purchased and a little of it will go a very long way in light use for knives.
