Even replacing the stock radio with another has not solved this problem:

The OEM radio's clock display goes very dim when the DRLs come on (i.e. when the engine is running and the handbrake is disengaged). It will stay that way until randomly, it will come back on. Then off again in a few minutes, etc, etc ad nauseum. Since replacing the radio did not fix the problem, I am suspecting a poor ground return wire somewhere.

Has anyone else had this problem? Thanks!

What's a DRL?
A poor ground, while possible, would not be my first choice. And you can check that out by creating your own ground... But I would think that while the clock and the radio use "different" power (12V) sources, the grounds would be the same... And you would probably see the effects with your radio as well...
It sounds like something could be loading down the circuit... From one of the DRL's?
Chuck

Thanks for your reply, Chuck. FYI, DRL = Daytime Running Lights

Haven't had a chance to trace this problem yet... will post back after I do.