I've confirmed my suspicion and provided a fix to my problem. I checked the isc-dhcp-server upstart configuration file:
and it did not depend on any interface before starting. I changed the "on start" line in /etc/init/isc-dhcp-server.conf to look like this:
I added the stuff after and including "and...". Now, after a reboot I get:
start on runlevel  and net-device-up IFACE=eth1
Now, please note that I'm running my DHCP server on eth1. Please adjust accordingly. Also note: it would be cleaner if the "on start" dependency automatically depended on any interfaces defined in /etc/default/isc-dhcp-server, but I'll leave that to ISC developers.
service isc-dhcp-server status
isc-dhcp-server start/running, process 2853