Encouraging people with the idea that "everyone is unique," while partially true, is but a futile attempt to claim a special niche within a world filled with billions of other humans. Though physically we are all different through DNA, and in personality through experiences of the past, we all fall into the category of human. And therefore not all of us, but rather none of us, are special at all. None of us can claim a special place amongst anyone else because we are all subject to the same human cycle: birth, life, and death.
There are names that have gone down in history moreso than others, of course. People of historical, creative, and religious importance. Presidents, writers, popes, etc. These are the names that large groups of people remember. And why? Because they did something significant; something that's lasted. Not because of who they were, but usually because of something they did. It is not their uniqueness of character that causes them to be remembered (though this is sometimes noted after they have achieved something great). It is their "uniqueness" of action.