It's not completely arbitrary.

There is a maximum defined by human usability: You as an individual don't want to wait for 24 hours to confirm that you've paid an individual unless you are dealing with an insanely large sum of money and checks. Even an hour feels long but would be tolerable. 

There is a minimum:
The minimum is defined by the need time for your node to get information, to process information, and send information.
Global network latency averages about 200 ms. Satellite latency can be longer at 800 ms.
This limits you to say 500 ms receive, block solving time, 500 ms send, so you can't have sub-second transactions easily without double spends and that's before you have to do any proof of work AND verification of transactions.

Since we now require a limit above a second, then what's a few more seconds?

Considering you want the method to be peer to peer, you need time to propagate between peers and peers in different networks and continents. The speed that you think the data can propagate between all peers now defines the next step: if you think you need to (synchronously) spread to 100 peers in order to reach thousands with the successful block, then you now are limited to a minute or two. So, a minute or two to just send and spread, and then a minute or two to receive. The delay because it's caused by the peer to peer action and verification. All before solving verifying other proof of work and solving the next proof of work.

Proof of work then becomes inconsequential. So, why not just double the time because ten minutes sounds good? The auto-adjustment every two weeks is arbitrary, so ten minutes is partially arbitrary because it has to work with normal behavior and try to account for changes in technology and interest.

Obviously there are faster ways to do things, but this was never just about speed. And those imagined network connections aren't completely real in practice.

There are other considerations, like with ten minutes you can do replace-by-transaction if you really feel you messed up and notice soon enough, but you also can't reverse it after that reasonable length of time.

So yes, it was partially arbitrary, but not completely.