Unknown coaxial cable female end. smaller than RCA or BNC. White rim We have a 10+ year old ASL MR safe long range optics eye tracking system that sends a video of the eye to a control unit. The eye camera output port is standard BNC, but the female (?) input on the control side is not. A cord extends from the unit with an end that is much smaller, resembling a 5V power supply connection more than BNC or RCA. The tip of the connector has a white rim.

The cord is annotated 75 ohm coax 1030 AWM 1378 200

The documented ("camera in", pg14) port on the control unit (at the other end of our mystery cord) looks proprietary.

Any idea what this connection is or what to use to adapt to BNC?
That's a barrel connector, as you say most often used for DC power connections. I doubt that it has an ideal impedance profile for 75 ohm video connections, but that probably doesn't matter in your case.

Get the inner and outer dimensions using a micrometer, match it to one of the standard sizes like type N etc, and you can easily build your own adaptor. I don't think you're going to find anything off the shelf since the use is proprietary / non-standard.