A white-label nameserver is basically your own nameserver, hosted by your hosting provider. It's commonly used for web hosting businesses that are on reseller packages, as it allows them to have "x.yourdomain.tld" as your nameserver, instead of using their own providers nameservers. This is done to give your customers the feeling that they're only dealing with you, and to help you establish your business as a "sole entity" if you will.
If our reseller customers didn't have this option, all of THEIR customers would need to point their domains to "ns1.mightweb.net" and "ns2.mightweb.net", which is obviously not ideal. Instead, our resellers can have their customers point their domains to their own, white-labeled nameservers.
EDIT:
After reading this when being slightly less tired - I'll modify the above slightly. What I described above is what I'd call private nameservers. Whitelabel actually refers to a nameserver that simply bares no resemblance to the owning brand. Ours, to give an example, are ns1/ns2.privatesitelabel.com. As you see, it in no way connects to our actual brand name (MightWeb).
On top of that, the WHOIS-details of said nameservers are private (by WHOIS guard/protection/whatever you want to call it) to avoid further linkage.
Of course, finding the owner of said domain is still rather simple if you get into it (tracking down the IP/hostname).