The best place to stay is going to be wherever they can find/afford a room, LOL. A large part of the rooms available for the game have been gone for months now, because there are many people who always go to spend the holiday in NOLA and to see the SEC champion play, whoever it might happen to be.
Geographically, the Dome is located on the northwestern edge of the neighborhood known as the CBD, about a mile up Canal St. from the Riverfront, right at the confluence of I-10 and US90. There is a large Hyatt right next to it, but that won't be available, because the SEC always books the entire thing for the game sponsor and the press.- If they follow past practice, the Florida team will stay at the Hilton Riverwalk at the foot of Canal street -- this is a HUGE hotel that traditionally is one of the last ones to have rooms available, but your folks might want to avoid drowning in that sea of orange and blue. (Yep, I checked -- that's where Florida is staying.)
Any hotel in the French Quarter, the Central Business District (i.e., the CBD), or the Convention Center area will be close enough to take a cab from. There really isn't much to do in the CBD when businesses are closed, so if they want to be able to walk around and people-watch when they are not at the game, I'd recommend the Quarter first and the Riverfront/Convention Center area second. (Tell them that if they want to stay in the Quarter, read the TripAdvisor reviews before booking -- the Quarter has some hotels that are VERY run-down.)
If they are driving, parking on New Year's is always at a premium, so they will need to allow for that. The airport hotels are now in an iffy neighborhood, as Kenner has gone downhill badly in recent years. Metairie or Slidell will do if they are willing to commute, but avoid New Orleans East if possible.