As a New Orleanian, the French Quarter is my least favorite area of the city. Stay in Mid-City, Garden District, or somewhere else in Uptown near Tulane. A St. Charles or Canal streetcar will bring you into the Quarter without a problem.
Renting a car is an absolute waste of money and effort (parking and whatnot) if you are (1) flying in, (2) staying in the city, and (3) not a baby about some walking. I live in the Warehouse district and got rid of my car a while back. I'm able to get around everywhere by foot, Uber, and street car/bus.
$5.95 is the minimum Uber fare (I doubt Lyft is much different)
Street car and buses are $1.25 per trip, or $3 for a one-day jazzy pass.