Is there any wearout mechanism for a relay simply from current flowing?
Yes, you can most likely use the mechanical lifetime for this purpose. However I have to question your premise a bit. 

The SSR (assuming it's an AC type, you don't say) will likely have an internal snubber so that there will be a small current surge at switching (depending on where in the AC cycle it switching) but it's unlikely it will reduce the lifetime by much. 

Still might be worth testing if this is an important point. 10^6 operations at 1 per second will only take a couple weeks.