This may be a coincidence, but I was farming uniques in Act 2 (using the unique farming ring) and did many many runs (50+), never saw The Black Lamp. Did all the variations of the genie/lamp based things on the map, still nothing.
I put on the Genie Ring for the first time and got the Lamp that run. Unless thats an incredible coincidence, it feels like it may be part of it dropping as a hidden effect. It makes sense if so.

If thats the case, is there any chance there are other uniques that have requirements like that?