Usually, a National Monument is designated by the President via Executive Order under the Antiquities Act. Some are later changed to National Parks via Congressional act. Sometimes Congress creates a National Park to begin with.National Park, National Monument, National Historic Site. Must be something to do with the funding.
Most National Monuments are administered by NPS, but as I recall some are Bureau of Land Management because they already owned the land prior to the Executive Order.
I'm not sure how a National Historic Site comes about.