There are several aspects about wizard names that remain unknown and will, in all probability, never be answered. Do the names have a symantic meaning relevant to their use? What is the method by which a name is assigned to a wizard? But there are aspects that we do understand and can explain.
From the moment a wizard is converted, they begin to develop a new name in association with their new powers. A new part of the name is given for each new tier that the wizard achieves. There does not appear to be any particular pattern to the names, but that may be because the names originate in some as of yet unknown language and given the right complexity of the language, it could take considerably more data than we currently have to infer a pattern.
What we do know is that, with each tier, the name that is given is longer than the last. This could be a representation of the power of that wizard, or it could merely be a manner of more specifically delineating their identity.
One concern that every wizard learns about is Name Stealing. The concern here is not so much a matter of danger against a wizard's person as it is one a concern akin to wizard identity theft.
When you learn a wizard's name, you can then use that name as a key to one of the abilities that specific wizard developed. When you learn someone's secondname, you can one of the spells they developed to gain firstname. Thirdname gives you an ability they developed to gain secondname. Fourthname allows you to use an ability they developed to gain thirdname.
Nobody has yet discovered a way to gain access to techniques developed for fourthname or after. Since nobody has ever gained a fifthname, abilities developed after gaining thirdname appear to be secure.
However, abilities used through name stealing have a lower potency than the original wizard. Also, once a wizard gains an ability through name stealing, they appear to become unable to develop that ability on their own. Because of this, most wizards only try to steal names of wizards who do not match their own type, that way they open roads for themselves rather than close them.
Magic performed through the use of name stealing resonate with the magic signature of the original wizard whose name was stolen. This is in contrast to what happens when magic is performed through soul stealing.