If a country wants to change its name, how can the new name be officialized? Should the
Country register the new name to any international organization? Is there any such organization that deals with the name change of a country? Or is it enough for the country to announce its new name to international society? What is the process that a country's
Name change is universalized and come into effect under international law?

The ruling body of that nation merely declares its new name following the requiremets of that nation. For the US since it is part of our Constitution, would or should require a constitutional amendment.

In a area where you have a dictator, I would assume he merely declares a new name.

There is no controlling factor of international law.