Could a bad hdmi cable be the cause of 60hz speaker hum?
From your description I don't think it's possible that it could be the source of the noise.

When you switch your receiver to select from a particular input channel, it would likely completely ignore or isolate any activity on any other input sources and they simply would not get into your mix at all.  The fact that you hear it on different things suggests that the noise is coming from a common source like you eluded to.

The easiest way to tell would just be to unplug the HDMI cable all together and see if you still get noise on RCA inputs if you really wanted to test the theory.  If it does magically get rid of your noise then I suppose we could start forming some theories about any power being transfered over the cable, but I would have doubts there because I've never heard of any significant power sourcing being done over HDMI.

HDMI carries digital audio (numbers with checksums), so it's more or less immune to noise that is not strong enough to flip bits or make them ambiguous to interpret on the receiving end.  If the data stream becomes corrupted, you would hear your audio drop out intermittently.  In any case, all of that is irrelevant because you're not even using HDMI.  It just happens to be plugged in and you've selected an analog source.

One thing you could try is to incrementally move your speakers closer to the AC and see if the noise gets stronger, or have your unit's electrical wiring checked out by an electrician.