Confusing oscilloscope calibration signal
From your description it sounds like there may be a tiny break in continuity somewhere or a dry joint, with enough capacitance present to couple the high frequency AC but ignore the DC.
Wiggling the cable whilst applying calibration signal may intermittently make it work properly again.

A simple way to test this would be too see if it works with a DC voltage (e.g. battery) at all. Or test for continuity with a multimeter.
If results indicate the worst, the biggest suspect is probably the BNC connector junction. If you can unscrew it (not moulded) then have a look and see if contacts are bad.