Barefoot_Monkey: Making a website where users can change their own name is often more trouble than it is worth, and it is relatively rare that a user would need a name-change, so GOG doesn't provide an interface for this. Having said that, it is most likely a simple task for someone with database access to change your name for you. Your best bet is to make a support request. My guess is that a friendly admin would be willing to change your name for you.
I once coded up a system that made it dead easy to change names without a user ever losing their unique identity. Surprisingly, it wasn't very difficult to do at all.
I agree it's probably not really needed -- and depending on how the GOG site was coded, may be near bloody impossible to implement -- but it can be done if you borrow the concept of the ID codes used on some image boards and apply that to a master account name that cannot be changed, while also providing a display name that can be changed.