A few oils and what they do:
Mineral Oil - non-drying oil, soaks into wood. Good for things like knives and cutting boards as mentioned.
Danish, Tung Oil - Drying oils. They're the ones that make oil-soaked rags light on fire. When exposed to air, they oxidize, harden, and become no longer oily. They're some of the common ones used for wood finish.
Linseed Oil - used as a part of the above two with additives, but comes in different flavors. Boiled drys faster. I don't know for sure, but I'm guessing that unboiled, which I've used as paint thinner, is more likely to damage other finishes and stuff. Check which one you have.
And definitely check on a hidden spot first, and probably wait a few days. Are you going to spot-finish them or do the whole thing? Very small spots shouldn't be too visible if something happens, but doing the whole thing would at least look consistent. Also, oil finishes are hand-rubbed. Apply, let soak ~30 minutes, then rub off with a dry rag. Mineral oil just soaks in and stays there, so doesn't get the nice hand-rubbed appearance.
Two other notes - I'm assuming these are speakers with real wood finish? Vinyl would probably melt. Also, oil finishes are generally not glossy much at all. If you want a glossy or semi-gloss finish, some oil-based polyurethane finish can be mixed in depending on how much you need. Oil has an advantage in that it soaks in, which the poly doesn't. The poly dries faster though and can be shiny (if you like that sort of thing), so mixing can provide a bit of both.
There are many other options to be found for people's special wood finish recipes. One involves melting some beeswax and mixing it with some danish and mineral oils or something. The beeswax I think helps a little with long-term waterproofing as well as adding another sort of glow to the finish.
*edit - saw you replied while typing the above.