Here was a cute idea I thought of. Divide the triangle into 4 by joining the midpoints (a "triforce" shape). The four triangles are congruent, each equal to a quarter of the larger triangle. Generate a uniform 1/4 probability event in your favourite way (flip two coins) and accordingly choose one of the triangles to contain your point. Repeat this indefinitely to get an exponentially small triangle containing your point.

I think this should be a uniform distribution by symmetry. Obviously the boundaries of the triangles never get picked but that's a measure 0 set.