What's the secret to finding the Farmer in No Man's Sky NEXT?

Answering my own question here, it turns out to have been, if not a glitch, at least a bad interaction due to a corner case.

What happened was that, after I had placed the Agricultural Station in my base, I was editing my base structure from outside. I briefly misplaced a structure so that it overwrote the Agricultural Station, but since I was outside, I didn't notice.

Apparently the result of this—since I'd already activated the Farmer part of the mission and it had told me to go find it—was that the Farmer showed up in the system, but jumped away from the place—station or planet—I was on so I couldn't reach it.

Once I went back inside my base (having given up, for now, on the Farmer), I noticed the Agricultural Station was missing.¹ I reconstructed it, went back to the station, and:

—

¹ This may be useful for some: if your inventory is filled in the process of deleting an element (because you have space in your inventory for the main item, but not subitems like glass for windows or you fill a stack when there's no slot in your exosuit to start another), you can build the Base Salvage Capsule, which is a large machine like a Large Refiner. It will contain any such items. (It can hold a huge amount, but allows only a one-way transfer to your exosuit.)

The mission is supposed to be quite easy. You are supposed to warp to the nearest Gek station, talk to the Gek there and find your Farmer.

There is no elaborate twist to the mission.

That being said, you seem to b experiencing a bug. Best bet would be to try to talk to every Gek on the station and to find out whether they can become your farmer or not. if that doesn't help, try another Gek station.