Microsoft has announced the addition of button mapping support to standard Xbox One controllers.
The support does carry a few limitations, when compared to Elite's options. The remapping support for standard controllers only gives you the option to swap button assignments, where as Elite allows the removal of an input as well as multiple buttons having the same input.
The configurations are also system-wide and saved only to the console they were modified. The support is specifically for Xbox One games at the moment, though PC support is expected to be confirmed soon.