Between the Staybridge and the IC, the location is pretty much a tossup. For a longer stay, I like the Staybridge for the extra space, the fridge to keep beverages cold, and the microwave to heat up any leftovers. What I like best about the IC are the balcony rooms; enjoying my coffee out there is a great way to start the morning - but these generally aren't available on award stays.
I haven't stayed at the Lemoyne, but I've been to their courtyard/pool area, it's a much more "New Orleans" feel than the others, and a better location if they will be hanging out in the Quarter.