Nice is expensive. Recommend instead staying in Villafranche sur mer. Quaint town on the beach, 15 min. away from center of Nice by train or bus. You can likely get yourself a lovely seaview room for what you'd pay for a hole in the wall in Nice.

AJ, as I was researching some answers to your question, I came across the following column from The New York Times, which is focused on budget travel options: http://nyti.ms/v7CxDr. In it, the writer recommends Nice's three-star Hotel La Petite Sirene. She also recommends visiting the Nice Convention and Visitors Bureau near the train station to get help and guidance with hotel bookings to suit your budget. Hope this helps!