I mean, when someone designs measure of, say, temperature, 0 means some important stuff: water freezing or molecules not moving. I suspect 12/24-hour day is an ancient invention, so 00:00/12:00/24:00 should mean some important time, but it's far away from dawn, sunset, zenith etc. What does it mean?
00:00 means "Astronomical midnight"
12:00 means "Astronomical noon"
If it be enquired, How can we know the Place of the Sun among the Stars, since all the Stars near it are lost in the Sun-Beams? 'Tis answered, that we can see plainly what Constellation or what Stars are upon the Meridian at Midnight, and we know the Stars which are exactly opposite to them, and these must be upon the Meridian (very nearly) the same Day at Noon; and thereby we know that the Sun at Noon is in the midst of them. So that when you have a Globe at hand on which the Stars are delineated, you find on what Degree of any Sign the Sun is on on a given Day, and see the Stars around it.
-- Isaac Watts, A.D. 1726
Source
Prior to time zones, 12:00 was the local solar zenith, and 0:00 the solar nadir.
The push for standardized times mainly came from railroads. A train schedule would have to indicate which city's time it ran on, and while the trains on one line might run on New York time, a different train would run on Philadelphia time, and if you had to transfer then you need to know how many minutes difference there was between the two times. A row of clocks in train stations would indicate the times in each city.