Well, true, the names of the months (and their existence) are pretty much arbitrary. But if you define a year to be the time it takes earth to revolve once around the sun, it makes sense to celebrate the end of one revolution at the end of the year. Why December 31? No idea, but my first guess would be that it's closely related to the winter solstice. We could've just as well defined the last day of the year to be the day of winter or summer solstice, for example. So yeah, the specific date is arbitrary.