Built-in styles, e.g., Normal, Heading 1, Heading 2..., cannot be renamed. What you're ending up with is what's called an "alias" for the style. For example, say I'd rather call Heading 1 and Heading 2 H1 and H2. By adding them as aliases, I can then apply these styles by, say, going to the Apply Styles pane and entering H1 or H2. (I don't use the Styles section of the Home ribbon, but I believe there may also be a way of getting styles there to appear by their alias, not their native name. No idea how to do this, though.)

If you want a style that is basically Normal, which, incidentally is a sort of "master" style, then you could make a copy of Normal and call it what you want. If you want that new style to be the same as the actual Normal style, then when creating the new style, select Normal in the "Style based on" field.