there are several timeshares in NOLA in and around the French Quarter, most are smal independent facilities in restored/renovated old buildings. Quarter House, Club la Pension, Jean La Fitte House, Maison Pierre La Fitte, Plaza Suites, Hotel de l'eau Vive are several that come to mind
Club la Pension was recently taken over by Blue Green and is now closed for renovation. They hope to re open for next years Mardi Gras
Wyndham has two larger resorts, La Belle Maison, and The Avenue Plaza Resort
As far as I know every timeshare resort sold certain event weeks, Mardi Gras, Jazz Fest and Sugar Bowl, generally. Unless you own these specific weeks you wont be able to get a reservation. The owners will usually use the week themselves or rent them out.
Wyndham is the exception they sell points. but unless you have advanced reservation priority you will have a problem getting in. I was on the Wyndham system when it opened up for standard reservations for Mardi Gras week (10 months ahead of Feb 17 2012, and there were a few vacancies. They were taken within 5 minutes
So long story short I think your best shot at Mardi Gras will be to rent from an owner that has snagged a reservation for that purpose. There is currently at least one ad here on TUG for next years Mardi Gras, and I know of two on Red Week. You could also watch Craigs list under vacation rentals in New Orleans
Mardi Gras is a wild time in New Orleans...and the celebration and parades start about 10 days ahead of Fat Tuesday. You might want to go the week of Feb 10, rather than Feb 17. especially with kids. I would take a hard look at Wyndhams Avenue Plaza. Its in the Garden District, not the French Quarter so things tend to be a little quieter at night, less flashers etc. (remember there is no closing time at the bars in the French Quarter...they drink and party all night).And several of the Parade routes pass right in front of the building. Parking is available cheap and the street car passes right in front.