What name will Shinigami Eyes show above the head of a person who changed their name? So let's say, Bob Schösel changes his name to Derkie Duckface. Would a person with shinigami eyes see Duckface's original name, or the name that is currently used?
Why would Gods of Death be concerned about a human institution of name registry? Within the TV series and manga, people go by aliases and those are never good enough. And there's 2 good reasons for this:


If simply changing your name changes the name that Shinigami eyes would see above you, then the value of having Shinigami eyes within the context of the story is greatly reduced, as there's no point in trying to find someone's real name. If changing your name changed the name the eyes see, then the goal of finding someone's "real" name would be changed in the story to finding someone's "current" name. Clearly from the story, the goal was to find the "real" name.
If changing your name changed the name that the Shinigami eyes saw, then it would be a way to circumvent death from the writing of your previous name in a Death Note. To play it safe, you could change your name every 5 seconds. There's not a rule to address this in the "How to Use" rules, so it follows that there's no such loophole with changing your name.


Therefore, the name you were given at birth must be the only name that matters.