What DHCP server are you using? Have you checked the logs on it?
Also, what is the DHCP lease time set to?
Is WinME patched up to date? Do you have any other machines running WinME without problems?
What other operating systems are running, connected to the LAN?
On the LAN, do you have NIC speeds set manually or automatically, as well as duplex settings? You should try manually as some cards can go berserk in some situations with autosense on. Laptop integrated NICs have a knack for this.
