The majority of the calendars on earth have used either lunar or solar cycles to keep track of time, ie days, months, years. What would a civilization cut off from the sun or moon use to keep track of time? Are there any geothermal or geomagnetic cycles that are regular enough for time keeping?
Biological clocks for cave dwellers are driven by hunger
In experiments on Somali Cavefish, which have been isolated from sunlight for million of years, the fish were found to be unresponsive to artificially imposed dark light cycles, but that their circadian rhythm could be modified by feeding them at certain time intervals during the day.
Other variables that can control circadian rhythms include temperature changes and social interaction. Most fish species in rest-activity cycle studies maintain a near-24 hour circadian rhythm; this is probably due to having been derived from a surface-dwelling fish and having no reason to change the activity cycle. However, one species in the last linked study changed their rhythm to 38 hours. The authors do not propose any specific reason for the change to a 38-hour schedule, but it is possible that it was driven by the food availability or social interaction mentioned in the previous studies.
Lastly, in the last linked study, the cave dwelling species that maintained a 24 hours cycle usually slept much less than their surface dwelling counterparts. Surface fish slept in 2-3 hour bouts for a total of 12+ hours. The cave fish slept for many 15 min or less segments for a total of only 2-3 hours.
A cave dwelling civilization organizes around communal feasts
For a species of cave dwellers with absolutely no contact with the surface and the sun, a likely result would be organize a time schedule around communal meals, ticking off both the hunger and social activity blocks for maintaining schedules. These schedules could be 24 hours, or any other time unit, apparently.
These creatures would probably not sleep continuously through a 'night' but would catnap throughout the day. Every 4 to 6 hours, perhaps, there would be a community meal, where everyone gets together and eats. For a stone-age level people, this meal would be regulated by the genetic circadian rhythm of each individual; the social interaction provide a means for calibrating the internal clock with the other members of the tribe. As technology advances, water-clocks or hour-glasses would begin to be used to help time the meals more precisely, until eventually mechanical clocks are invented.
It would be good to note here that the way we currently keep track of time is completely arbitrary in the first place. We define seconds, minutes, hours, years and so on, but the truth is that any definition would make just as much sense as long as it was consistently applied throughout your underground society.
For the sake of convenience, let's assume that they use the same units with the same hierarchy as we do. Assuming also that these creatures have 10 digits, and thus use a decimal numbering system, it would seem to make sense that their units of time follow a decimal system too - i.e. 100 seconds = 1 minute, 100 minutes = 1 hour, and so on.
In that case, the main question here would be: what consistent, reliable natural pattern would they use as the basis for their second? Since most geological phenomena happen pretty slowly, these would not be great for timekeeping. Instead, consider that your society, at some point, developed The Pendulum. Perhaps some great thinker saw a large block of stone tangled up in an underground creeper, swinging all by its lonesome, and he had an " eureka" moment.
I imagine a small group of Watchers (pun intented), people whose job is to monitor The Pendulum and to reset it to some initial state at regular intervals (for instance, every 10,000 periods) so that the rest of the society may continue ticking along.
The concept of days and years was forced onto humans due to the existence of natural cycles we have no control over. Things like the sleep cycle and impact of the seasons. Our existing calendar is a mess mathematically, we have to somehow reconcile the length of a day with the length of a year which don't factor nicely, hence hacks like leap years.
Underground though with no seasons there is no such problem. The need to organise activity around the sleep cycle would probably require some kind of "day" unit being created to synchronise society. After-all you can't organise a meeting by saying "lets meet at sunrise or noon" in that case. The need for some kind of clock system to organise future events would probably be a lot more pressing. I can imagine people scurrying around the caverns whose sole purpose is to synchronise sand timers.
Once they have a day unit they don't need anything more. They can measure time just in number of days. Any official calendar created with longer periods would probably be far more logical, say base 10. Or they might use the age of their current king in days to denote historical events.
Any faint geological cycle, or even some sort of impression from the surface - perhaps some kind of bat species migrates down in accordance to season at the surface the humans cannot see - I doubt it would affect their calendar, such things would just be of some academic interest to a few mathematicians who are the equivalent of astronomers (or astrologers), who would look at the cycles to try and find meaning and generally be frustrated at how they aren't predictable and don't align properly with their measure of a day.
The earths rotation would still work. Pendulum deviation could be measured. once they built long pendulums for any reason they will notice it.
Foucault pendulums in particular are particularly noticeable.
Timekeeping is even easier. Simple sand glasses, water clocks, and other escapement based measurements still work fine as well. The first of these did not match known cycles they just made them tailor made for particular purposes.
Edit: dumped the tidal force idea, possible but too unlikely to be noticed in the first place.