What is this circular rod with wired wrapped 'coil' and ceramic capacitor?
I'm using 3 pieces of information to suggest a hypothesis as to the function of that device:


It's an LC tank circuit, and so will have an RF resonant frequency (which could be calculated from the values of the inductor and capacitor).
This LC circuit is not connected to anything and so its function must be self-contained, not part of the equipment it was found inside.
It was effectively hidden, by being in the battery compartment (of whatever this equipment is).


Hypothesis:

Perhaps it's a device which allows the device to be detected as it passes a suitable RF transmitter e.g. as used by some shoplifting detector systems, hire shops &amp; even office asset detectors?

Although current systems using that technique can have flat stickers with the L and C formed internally, they use the same basic technique of an LC tank circuit as in your photo, whose presence can be detected when it is "excited" by a transmitter tuned to the LC circuit's resonant frequency. Yours could be a home-made equivalent, or an early version of that technology.

Whatever the equipment where you found it actually is, that LC circuit would allow someone to detect when it passed a certain point (where a suitably tuned transmitter was fitted).