Minint stands for Minimum Windows NT - It is a preinstallation name used by things like WinPE (Windows PE) or even BartPE. The alphanumeric characters are the serial number of the system. This is found in the bios, so when WinPE or another program like it run, it pulls Minint and adds the unique serial number to the name.
First you need to know how they are getting there. I'm guessing you have dynamic updates turned off, otherwise when the PC picked up a new name with the same IP, the system would update DNS. As it stands it sounds like dynamic updates are off, thus only the DHCP server can update DNS.
That being said, there is a chance you are updating your environment in a non-standard way. An example being dumping the ARP tables from your switches into DNS. That's not a popular way of doing it though.
Verify if you have dynamic updates allowed on your DNS server.