So I was having a discussion about angles with a student today and they were given a problem like:
convert 3$^{\circ}$ 15' 24" into degrees which is straightforward enough,
$3 + \frac{15}{60} + \frac{24}{3600}$
Afterward the student asked the natural question "Why do we call these minutes and seconds?"
This led us into a discussion of what a "degree" is in the first place. Basically the idea is that the Earth will travel through very close to 1 degree of its orbit in one day since there are 365 days in a year and 360 degrees in a rotation.
Once that is understood we can easily see the connection between angles and time. The question that occurred (to me not the student) after this discussion was: Wait, what about the hours? If a degree is roughly the angle the Earth travels through in a day, then shouldn't a "minute" the the angle that it travels through in a minute of time? But this is incorrect because we are seemingly missing the "hour" measurement.
I imagine that there is some historical reason for this, does anyone have any insight?
We really don't know for certain where the $360^\circ$ convention comes from. But, the best guess is that it is Babylonian.
The minutes:seconds convention is definitely Babylonian. The Babylonians used a "Sexagesimal" (base 60) numbering system.
1 minute means $\frac 1{60}$ and it could be applied to time, or distance or degrees. 1 second means $(\frac 1{60})(\frac 1{60})$ Rahul provides some more history below.
24 hours in a day (broken into two 12 hour halves) and 12 months in the year, seems to have come to us from Egypt.
Minutes were not used for time keeping until we had sufficiently accurate clocks (late middle ages). So, the minute was a unit of angular measure, long before it was a unit of time.
And seconds as a unit of time came along in the Enlightenment (1700's).
Base 10 numbers didn't make it into Europe at least until about the turn of the millennium. And base 12, base 20, etc. can still be found in some of our archaic measurements. (dozens, scores, shillings, inches, gallons, etc.)
360 has one more thing going for it. It has a lot of factors. Which if you lived in the world before the invention of the decimal point, is probably a nice characteristic.
It is quite possible the the 1 day $\approx$ 1 degree is a happy accident.
However, beyond high school trigonometry, the degrees convention is out of favor. Measurements in degrees does not play well with calculus. So all "higher math" will use radians, instead.