There's not really a whole heap you can do, especially not without knowing what is behind your dhcp and dns. obviosuly it's a *huge* security issue if someone in your position could put a box on the network and have the name of your choosing suddenly being associated with that box. what if you choose a name that's used elsewhere? you could take over the main mail server name etc...
When a DHCP request is made by a server it will often pass along the hostname in the request for reference. often ip addresses are assigned by hostname explicitly, as opposed to by mac address, or indescriminantly from a pool. as you got an ip address on dhcp after doing nothign centrally you clearly have an open DHCP pool to pull from. i would *guess* that the orginial box pulled a lease down from DHCP and a DHCP guy made that address reserved for that MAC address. and the DNS name was stuck to that IP one way or another.
what happens if you ping the hostname you are after? does it resolve to the old address still? if so i would suggest that you should possibly look at a mac spoof on the new box so the DHCP requests appear to be coming from the old addres - OR - take the nic out of the old machine and reuse it, if possible. if you want to try a spoof, check here:
http://whoozoo.co.uk/mac-spoof-linux.htm but there are still other issues that would affect how the DHCP server assigns the address. not least you already have a lease, so by default your box will ask to keep the same address regardless. you can probably delete the lease file which tends to live somewhere like /var/spool/dhcp/leases or similar.
one last thing is to just stop using DHCP and use the old IP address. if this address is based on DHCP though the lease may well expire and be dished out to somethign else on the network too, which would be nasty.
these are most definitely hacks and workarounds though.... nothign beats talking to the sysadmins and getting their view of things.