Writing Hostname to router

My D-Link DI-624 supports "static DHCP" which serves my small network needs
well. The router setup page shows a table of preset ip addresses and the
hostnames they belong to. The router doesn't seem to use the hostnames
itself (e.g., for DNS), they're just for user convenience.

Problem: I regularly use SystemRescueCD for maintenance and backup on each
host ... and the static DHCP table hostnames get changed to sysresccd. All
of them. AAAArgh. I recall trying a VectorLinux live CD that did the same
thing.

How do I teach Debian Sid to do the same thing: update the router's table
with the (real) hostname of the Debian install I'm running? This would
fix the sysresccd mischief and make me happy.

Thanks!