Between the student apartment and Clara’s AirBnb, I had the chance to stay at one of the hostels offered by Bureau des Voyages de la Jeunesse. They have quite a few locations, from the Louvre, to Champs-Elysses, and even one in the Latin Quarter. I chose Opéra/Montmartre for its proximity to Palais Garnier (and also because their lobby looked really cool).
I booked directly through their website, which I recommend--even only two weeks ahead of time, the prices were fair and much better than third party websites. Checking in was really easy and I had a room with a few other american girls (you can choose whether to room co-ed or not). There were elementary age kids to college professors, and people from any variety of countries. Breakfast is free, wifi is paid nightly, and basic amenities (towels, small toiletries) were available at the front desk. And the hostel itself is beautiful (just loook)
This tiny traveler rates it 5/5 (don’t pass this experience up)!