The cheapest time to stay in New Orleans is usually during the summer, between June and August. During those months, the weather is very hot and humid, discouraging some travelers from visiting. August is the start of hurricane season in New Orleans, so it may be rainy for days on end. All seasons in New Orleans have their pluses and minuses so it really comes down to whether you prioritize cost, good weather, small crowds, and so on.
However, you may also be able to find lower-than-average prices in January and February (outside of Mardi Gras), or in November and December before the holiday rush begins.