I think they found it via some random port scans, perhaps. There are other indications it may have been scanned as part of a list of Bitcoin daemons that were online or relay nodes, yes, but that's harder to determine. Looking at the attack pattern once they found a way to get through, they were browsing around and seem to have possibly taken all the coins as a sudden opportunity, and maybe not the original goal. It appears they did a little more looking around on the filesystem after they transferred the coins. They never actually stole the wallet itself, as far as I can tell, though I'm still reviewing traffic flows.
I'm hoping the person has some sort of conscience in all this and returns them, which would be ideal, but in reality, the traces left behind seem very amateur.
And no, not running Java as a browser plugin or anything. The compromised system isn't used as a desktop.