It's an Atlas #200 Snap-Relay. It's an electrically operated double-pole double-throw switch. Oil would be bad for it. I'm wondering if I can puff some graphite in there. That'll take care of the lubrication, but if too much goes where it's not wanted, it could also short out the electrical contacts.
The Snap-Relays are mounted upside-down under the layout. I took some Greas-em, held the applicator pointing up into the place where the manual actuator is, and gave it a little puff of graphite. It worked without shorting out the contacts. Most likely the secret is to use it very sparingly.