I used to work for a company that had no internal DNS, DHCP or a consistent hardware process. Fun times. I feel your pain.
This situation is very similar to a programming language: you write a piece of code that does something and then you re-use the code. You do this because repeating the same task over and over is not cost effective. You also have the issue of updating all the instances of said code when a change is made, or a bug is found, etc. The same applies here.
Firstly, point out the central point of administration an internal DNS server would bring. An update in one place would bring joy around the entire network.
Secondly, make them aware of the bandwidth savings. Think about it, what DNS settings are being pushed out by DHCP? It would have to be some sort of external service. Having an internal server would eliminate the outbound round-trip which would speed things up for your internal users and decrease the bandwidth usage on your externally facing link. Win!
Security/control! You can black list known malicious DNS entries on your DNS server - i.e. you can have your DNS server resolve nasty-website-that-is-evil.tld to an internal page that explains why it's blocked (OK, this can be done with a proxy server as well, but there you go).
The list goes on and on, really.
I hope you've now got something to go by.