If the host network supports this, of course DHCP is the easiest way to provide an appropriate IP address to an embedded box.
Additional Hardware could be used: a serial flash or dip switches.
Maybe the Dimm-PC has a CMOS RAM or a parameter Flash for the BIOS. Here some bytes might be available.
You _could_ put the information in the BIOS code FLASH (if this is a separate device) but who would dare to recommend this ?
Additional to the compressed file system, the Linux boot information contains the Kernel command line. Supposedly it's easier to change this than the internals of a compressed file. I don't know how to request information off the Kernel command line from a user land program, but I do believe that this is possible, at least when writing a device driver for that purpose.
You could reserve an additional section in the flash Linux boots from and access same directly (e.g. as a memory location) with a device driver.
-Michael