You are remembering the ergodic hypothesis in thermodynamics, whereI think the name originated and then was generalized.
Is there a relationship?
In a sense. Ergodic hypothesis is basically that the phase space is explored "equally" (in a precise sense), over time. Ergodic random processes are pretty much independent of their initial state. So at a hand-wavy level its a very similar idea.
Is that not the right heuristic?
Also -- why is it called ergodicity? I keep thinking it has something to do with work or thermodynamics, but I'm missing it...