Well obviously not completely arbitrary - it's entirely based on earth's magnetic field.
As for which way we call 'up' - here's an excerpt from the 'North' Wikipedia article as for why 'North' is up instead of 'South':
Up is a metaphor for north. The notion that north should always be up and east at the right was established by the Greek astronomer Ptolemy. The historian Daniel Boorstin suggests that perhaps this was because the better-known places in his world were in the northern hemisphere, and on a flat map these were most convenient for study if they were in the upper right-hand corner.
Edit: As user javindo points out, the excerpt is especially true for globes where studying land on the bottom of a globe is especially cumbersome. It's hardly arbitrary that ~3/4 of land is in the Northern Hemisphere.
As for which way we call 'up' - here's an excerpt from the 'North' Wikipedia article as for why 'North' is up instead of 'South':
Up is a metaphor for north. The notion that north should always be up and east at the right was established by the Greek astronomer Ptolemy. The historian Daniel Boorstin suggests that perhaps this was because the better-known places in his world were in the northern hemisphere, and on a flat map these were most convenient for study if they were in the upper right-hand corner.
Edit: As user javindo points out, the excerpt is especially true for globes where studying land on the bottom of a globe is especially cumbersome. It's hardly arbitrary that ~3/4 of land is in the Northern Hemisphere.