Before king Charles V decided to choose the Louvre as the official Royal Residence in the 15th century, it used to be the Royal Palace. When the king left, he left a keeper to look after the building : his name was the "concierge" and the name "Conciergerie" first designated his personal house, then the appellation was extended to the rest of the building and, during the Revolution, it referred to the prison where the aristocrats were waiting to be beheaded... In the same way that the Concierge kept the keys of the Palace for the king, the Concierge of your hotel will keep your key while you're off with me!
Now, to be honest, I wouldn't put the Conciergerie at the top of your list, especially since it's been restored, modernized, computerized and sanitized by "official-people-who-know-better" in 2016 : to me and many other guides, it lost a lot of its sad emotional charm... but if you go to Sainte-Chapelle next door, it will cost you just a little supplement to go inside (and there's no line), and you'll get to see a beautiful big room of the early 14th century considered one of the most beautiful Gothic rooms of secular architecture in France.
The Conciergerie is also a good opportunity to talk about the last sad two and half months of the life of Marie-Antoinette (as she was emprisoned there) but, if you are interested in her and the troubled times of the French Revolution, you should go to the Carnavalet museum (when it reopens in 2019).