Out of the box, when debian/sarge requests a DHCP address, it doesn’t send up a hostname for a dynamic DNS server to use.
It is simple, but seemingly different on each distribution. But for debian/sarge, create a file /etc/dhclient.conf and put this in it:
send host-name "yourhostname";
Of course, you can put other options for dhclient in there too.
Looking for how to do this for Red Hat Enterprise Linux? Look here