My wooden cuttingboard also must be oiled periodically—quite a bit when new, but much less now. I get a food grade mineral oil from Sur La Table. It lasts a long time. My thinking is that it is so refined, it can’t go rancid? Only a guess, but I’ve never had any trouble with rancidity or odors.
That said, I’ve also always heard that a rolling pin should only be wiped off, also. That said, I don’t think mineal oil would bother it, considering my experience with the cutting board.