The star- month is what modern astronomers call the sidereal month, that is, the average time it takes the moon to return to the same background of stars. 

Early Egyptian astronomers are likely to have known quite well how long the moon needed for that journey.  They believed it to be a god, so they must have carefully recorded how many days passed between major events in its phases and path; their next logical step would have been to accumulate those records to compute averages for periodic phenomena.

Already in pre-dynastic times, even before their adoption of the 365-day civil year, the Egyptians used a lunar calendar.  You can see this attested by the hieroglyph for "month" which depicts a crescent moon above a star[1].

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Later scribes often omitted that star and replaced its image under the crescent with the number of the month.  However, the moon sickle with the star remained the formal sign for the 30-day months of the civil year although these had lost any connection with the moon.

The Egyptians kept this ancient lunar calendar in parallel with the civil calendar until the end of pharaonic times.  They used it as a liturgical year for determining seasonal festivals[2], just as we use in parallel with the months and years a calendar of weeks that is independent of both and also determines many of the religious festival days.

Unlike the 365-day solar- based civil calendar, this lunar counterpart had leap months to keep time in tune.  Its designers had named the twelfth lunar month for the rising of the bright star Sirius, and to keep it in place against that event they introduced an intercalary month whenever it fell into the last eleven days of the Sirius month.  This happened every two to three years.  

The ancient Egyptians were thus quite familiar with the idea of leap periods but somehow did not transfer it to the civil calendar, just as they had mastered the building of vaults in brick from at least the first dynasty on but did not transfer the technology to stone until more than two millennia later, during the Twenty-fifth Dynasty in about 750 BCE[3].

The fact that the "month" hieroglyph shows the moon with a star attests that those ancients followed not only the phases of the moon but also its motion among the stars.  Indeed, they had divided a band of stars into 36 sections called “decans” that succeeded each other in ten-day intervals, as attested by some elaborate “star-clocks” on coffin lids and tomb ceilings, and also by the mention of “night hours” in the Pyramid Texts[4] from about 2,300 BCE.  

This mapping of large sky areas allowed the ancient Egyptians to tell time at night[5] and also to plot the wanderings of the moon quite precisely.  Watching the moon sickle inch up to a star and then suddenly occult it with the dark part of its circle makes a dramatic spectacle and makes it easy to pick a spot in the lunar path from which to measure its course.  

In just a few years, a steady observer could easily have determined the period for the moon's return to that spot with at least the precision of about 35 minutes per month that is the difference between the ratio on the mace and the modern value of  27.3217 days. 

That value is likely to have been similar in Narmer's time.  However, modern astronomers cannot extrapolate it so far back with the same four- digit precision, so we cannot evaluate just how close the 2.73462 ratio from Narmer's " booty list" came to the actual value back then.  Moreover, the other constraints on the entries in this mathemagical composition could have obliged the Narmer artist to accept some compromises in the closeness of the hits.  

In any case, the mace ratio's difference of only 0.0897% from the modern length of the star month is consistent with the ancient measurements of that month which the astronomer Gerald Hawkins deduced from a stela dated to the reign of Ramesses II (1290 to 1224 BCE).

The inscription on that stela tells a lengthy story about a princess in the mythological land of Bekhten. Her health is waning from spirit possession, and the Egyptian moon god Khonsu travels back and forth to heal her.  

The text gives a great many precise dates and durations for those trips, and Hawkins showed that they amount to a detailed account of the moon’s travels along the horizon during the 18.6-year cycle of the lunar nodes around its orbit.  The length of the sidereal month embedded in this narrative differs by 0.0143%  from the modern value[6].  This is even closer than the corresponding ratio on Narmer's mace from almost two millennia earlier. 

With all the evidence for the ancient Egyptians' keen interest in the motions of the moon, it seems more likely than not that the mace designer and his priestly colleagues knew a fairly good approximation to the length of the star month and could have incorporated it into their web of number nodes if they wanted to.

The same is true for the length of the 365- day civil year: