I can imagine in prehistorical times keeping track of midrange time by the phase of the moon, if months were kept synced to the moon we would probably name weeks as such, waxing gibbous, waning crescent...
On the subject of keeping months in sync, I can imagine having a fun festival season around the weird 13th month(all months being 29.5 days long) that bisects the new year.
The tricky part about months (and weeks by inheritance) is that while the moon phases are a nice easy mid-range timekeeping device(something between days and years) they don't line up with days or years very well. So as your society settles on having 4 important religious observations per month and starts keeping better time. It changes to having that observation every 7 days and delinking the month from the moon so it does not drift around the year so much.
I don't really know how the BCE biblical world did the week keeping. but our modern interpretation was mostly solidified by incorporating approximations of that into the other big influence on Europeans society, the Romans. Who used a month and half month system so it fit fairly well.
24ths and 60ths are useful fractions because 24 and 60 are highly composite numbers, i.e. they have many divisors. 60 is a particularly good choice, since it's a "superior highly composite" number; and so is 12, if we consider an AM or PM period, rather than a full rotation.
Such divisibility was important before the invention/discovery of general fractions (i.e. the rational numbers). Babylonian arithmetic used something akin to base 60 floating-point, which could represent halves, thirds, quarters, fifths, and their sums and products (but not e.g. sevenths, or reciprocals of higher primes; in the same way base 10 floating-point can't represent a third).
Minute, meaning "small", describes the shifting of the radix point in such a base 60 floating-point system. Second, as in the ordinal 2nd, describes a further shift of radix point; and likewise for thirds, fourths, etc. for ever-smaller increments.
One of my favorite watches for weekends and vacation is a single-hand 24-hour watch.
The solar watch face on the Apple Watch would be a great implementation of the 24-hour dial, except it doesn’t let you turn off the more precise time in the center.