A high pitched tone ( probably 60 Hz) is coming from my electric water heater's upper thermostat. We have plenty of hot water, the pressure is normal, I've drained the tank, and flushed the pressure relief valve, this is not a water gurgling noise.

I adjusted the temperature setting slightly lower. The tone stopped for a few days, them started up again. When I turn off the water heater's circuit breaker, then turn it on again, the tone stops for a while.

The water heater is about 8yrs old. I suspect that the thermostat is oscillating. The tone is annoying, but otherwise the water heater is working fine. Is there something I can do to stop the noise, before I replace the thermostat?

All you can do is replace that thermostat. It's old and should have failed before now, I would estimate.