This is my elderly sister's car. There is a clicking sound under the hood and her mechanic cannot find the source. Replaced alternator and belt. Is there a sensor or bearing that would make that sound and not show up on a diagnostic check?

There are several possibilities.... I remove the drive belt then start engine to see if noise is gone.. If so, then it is in the drive belt//components.. From that point the componets can be tested by turning each by hand to find the culprit.. If noise is still present with belt removed, then further testing with automotive stethoscope should find culprit..... Test, don't guess!

The noise that I have found in which case you are hearing is the clutch on the A/C compressor. The noise wont hurt anything. I figured it out 20k+ miles ago. More than likely because of its age.