We checked with a couple of experienced Extension specialists, and they suggested a "heavier" oil type (as in thicker, not thinner), to avoid having the oil layer dry out / evaporate over time in storage, thus rendering its corrosion and lubrication protection less effective. One person used "40 or 50 weight fossil fuel oil." If you were to use a thinner, faster-degrading oil type, then we presume the trade-off is that you'd just need to reapply it more often. None of our respondents were concerned with the small amount of oil left on the tool surfaces being a significant risk for plant damage (no one observed damage from their own oiled tools), and we don't think it would attract insects to the storage area, so that's probably not a reason to avoid any particular oil type by itself.

Miri