I was searching in the internet to find the reason each degree equals to 60 minutes and I found this:
60 seconds make up a minute. One day is equal to 24 hours, and the hour hand completes two 360-degree circles around the clock in a single day, for a total of 720 degrees. One degree (2 x 360)/12 equals one hour.
But does it make sense? Why should we divide it by 12? An hour itself equals 30 degrees. So each second equals 6 degrees and thus, each degree should equal 1/6 seconds.
The terms "minute" and "second" were likely more commonly used to refer to fractions of a degree than for time back when they were invented. The meaning and pronunciation have slightly morphed over the years and are generally synonymous with time now.
But if you think of the homographs of the words (words that are spelled the same, but have a different meaning), minute pronounced in the way that rhymes with "my newt" just means "very tiny". So a "minute of a degree" just means a "tiny fraction of a degree", or more specifically "minute division".
Similarly the homograph of "second", as in "not first", "not third", but $2^{nd}$. So a "second of a degree", just means "second division".
Before the age of computing machines, division was time consuming, so it made sense to choose a base which was easily divisible by many numbers, and 60 fits that quite well, and is the basis for the sexagesimal system. And we still use this system today in that a "minute part of an hour" is 1/60th of an hour, and the "second division of an hour" is 1/60th of a minute.
So the meaning of the words still holds today, in that they mean 1/60th of something. It's just that what they are 1/60th of changes depending on context.
In astronomy, the words arcminute and arcsecond are generally used to disambiguate the terms. But in celestial navigation and land surveying, it is still quite common to just use minute and second to refer to fractions of a degree.
The history section of the Wikipedia entry on the minute contains some of this explanation, and expands on to "thirds" and "fourths".
Historical reasons.
Yet your base assumption that 24 hours or 360° equal two revolutions is wrong in an astronomical sense: An hour equals approximately 15 degrees of movement of stars or the Sun if you are dealing with astronomy. Yet that's inconsequential here:
60 is nicely divisible by 2,3,4,5,6,10,12,15, and 30. So subdivisions of degree were chosen with base-12 in mind, making coarse calculations without floating point numbers easy and also marking easily 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/12, 1/15, 1/30 on geometric tools. And the same holds true for dividing each minute into 60 seconds. On the same reasoning the full circle has 360° degrees... nicely divisible without leftover by many numbers, too, introduced this way already thousands of years ago in mesopotamia by the Sumerians and Bablyonians. See also this answer in the math SE or the wiki article on the sexagesimal (60-base) system.
Generally, in former times quantities easily divisible were traded, especially of goods where the unit 1 is not easily split, like a complete oxen or eggs. Eggs are here often sold in units of 6, 10 or 12.
While the traditional unit systems usually are a mess and PITA, they sometimes use the factor 12 as well for the conversion between different units like between foot and inch (but not others; just look at the funky conversions between e.g. fingers, yards, miles, furlongs or queen's chains).
One key side effect of the French Revolution was the introduction of the metric system. Prior to that revolution, how length, area, volume, mass, and money were measured varied not only from country to country but from town to town. It was an absolute mess. The metric system addressed all of these.
Two things that people measured were not an absolute mess: time and angle. With regard to time, the scientific community had universally settled on 24 hours per day, 60 minutes per hour, and 60 seconds per minute long before the French Revolution. The scientific community had also settled on 360 degrees of arc per rotation, 60 minutes of arc per degree, and 60 seconds of arc per arc minute. These measures date back to ancient Egypt and Babylonia, so very, very old.
The French Revolution's attempts at metricizing time and angle failed because these were the well established de facto standards. Were the goofy standards? Arguably so, but the standards did exist. That there were no standards (de facto or official) for length, area, volume, or mass were key in the rapid advancement of the metric system -- except for time and angle.
The Earth does not turn through a minute of arc or an arcminute during a minute of time, nor does it turn through a second of arc or an arcsecond during a second of time.
Assuming that ther is a connection between arcminutes and minutes of time is an error.
In angular measurement:
A circle is divided into 360 arc degrees, each arc degree being 1/360, or 0.002777, of a circle.
Each arc degree is divided into 60 arcminutes, each arc minute being 1/(360-X 60) or 1/21,600 or 0.000046296, of a circle.
Each arcminute is divided into 60 arcseconds, each arcsecond being 1/(360 X 60 X 60), or 1/1,296,000, or 0.000000771, of a circle.
In time measurement:
One day is divided into 24 hours, each being 1/24, or 0.0416666, of a day.
One hour is divided into 60 minutes of time, each minute of time being 1/(24 X 60), or 1/1,440, or 0.000694444, of a day.
One minute of time is divided into 60 seconds of time, each second of time being 1/(24 X 60 X 60), or 1/96,400, or 0.000011574, of a day.
The Earth turns 360 degrees of arc in one 24 hour day of time.
So the Earth turns through 15 degrees of arc in one hour of time.
And thus the Earth turns through 15 arcminutes during one minute of time.
And so the Earth turns through 15 arcseconds in one second of time.
So any source implying that an arcminute is equal to the angle that Earth turns in a minute of time is incorrect.
The Earth does not turn through one arcminute in one minute of time, nor through one arcsecond in one second of time. So a different number of arcminutes fits into a full circle than the number of minutes of time during a day.
