I figured it out. So, essentially usb0 needs to be configured as a network interface. I followed these steps to enable it.
1) add the following line at the end of your /etc/network/interfaces file:
iface usb0 inet dhcp
2) sudo ifup usb0
3) sudo ifconfig usb0 192.168.2.2
After this when you do ifconfig you should see an entry for usb0 with an inet addr:192.168.2.2. You can then connect to the Edison via ssh firstname.lastname@example.org. Apparently the IP address 192.168.2.15 gets assigned by default to the Edison.
A slightly easier way is to put all of the usb0 interface information in the /etc/network/interfaces file. The following stanza is sufficient.
iface usb0 inet static
Note that you should use static IP assignment rather than DHCP (unless you have something on the 192.168.2.x network providing DHCP). Once you have this information in your interfaces file, simply running ifup usb0 will configure everything properly to be able to talk to the Edison over the USB network interface.