Not what I would recommend, Mineral Oil.
I use Sewing Machine Oil, made for machinery that gets hot, with FINE clearances needed EACH stitch. Have not found Sewing Oil to get sticky or gummy for me. Sewing Oil is pretty available, always 40% off on sale or with a coupon at JoAnne Fabric stores.
I store the oiled-after-use clippers in a sock (which gets oily) to prevent moisture reaching clippers and blade dulling if they get moved around between uses. They are then put inside a canvas bag, with the sock on clippers, when put away in the closed tackroom. No rust or moisture issues with this method in our humid location. Clippers are never left out, used, cleaned, put away. Some are pretty old!! Still work well and look nice.
Extra Blades (not attached) are kept in a locked plastic box to keep moisture out. I usually add the extra dessicant packs received in new shoe boxes, new bottles of pills, which absorb moisture. Seems to work pretty well, no blades get rusty in storage boxes. They come back from sharpening in those little brown envelopes of paper, marked with sizes. I do think that helps with keeping blades sharp, since paper glue lets go to tell moisture has gotten in. I have a LOT of spare blades, various sizes, so I invested in good Snapware storage (JoAnnes with coupons again!) boxes. Even with the bargin price of blades at a tack sale (A5 for $1 a set!), the cost of resharpening ($5 the set) makes it well worth the time to store them well.