(1) My speculation is that the number 60 has some kind of religious or cultural significance. For example how the number 9 is significant in many tribes, as it's the approximate time the baby is in the womb. It doesn't seem to be more convenient than 10 at a first glance. The only difference, mathematically, between 60 and 10 is that 60 can be divided by 3.
(2) We use 60 when measuring time. There are 60 seconds in a minute, 60 minutes in an hour, etc. To number of angles at the center of the circle is 360 degrees, which is 60x60. I am unfamiliar with the Chinese zodiac, and I am not sure where else we'd use 60.
(3) From just reading on the web (I couldn't find a legitimate source), I've found that the number sixty is a "superior highly_composite number" (https://en.wikipedia.org/wiki/Superior_highly_composite_number). This means that sixty is divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Therefore, the reason why the Babylonians might have used it is because it's easy to work with fractions, if we make the common denominator 60 or 3600.
Good post Marius. The significance of numbers within systems of religion, faith or worldviews is a fascinating topic to explore in the context of histories of mathematics.
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