The issue was a DNS problem on the client's network. When using the fully qualified server name, both in the Mid-tier config and the Admin/User tools, the flashboards show up fine.
As a side note, one of the things I tested was logging in with the ip address of the server. Although flashboards would show up then as well, it was taking a few minutes for them to come up; yes, a few minutes.
It would be nice to get an error that was more accurate to the actual issue though. I wouldn't think that one would normally look at DNS issues when the ARERR was 3624 (invalid password). That error seems to indicate you're hitting the server, just not authenticating. In fact, the arfbserver.log file showed that the variables were being parsed, which again seems to indicate the flashboards server is communicating with ARS.
my 2 cents