I would suggest staying in the Central Business District just adjacent to the French Quarter on the other side of Canal Street.  Omni Royal Orleans (note: there's one in the CBD and another in the French Quarter) and Loft 523 are growing favorites, and Le Pavillon Hotel is great (hot chocolate and PB&J happy hour!).  One hotel that I would avoid is Hotel Modern.  It's a fairly new hotel, so there will probably be some buzz about it when you search for NOLA hotels, but I was very underwhelmed by my stay there.

If you want to stay more towards the Frenchmen Street side of things, Le Richelieu has high ratings.

One word of advice should you ever find yourself stuck in a hotel on Bourbon Street again: request a hotel room that faces the pool, or the center of the hotel room, and not the street.  Unless, of course, you enjoy throwing beads at passerbys on the street from your balcony :P

I stayed in the St. James Hotel on Magazine Street, which was a few blocks from Jackson Square but far enough to stay away from the craziness. It was quiet and had a great restaurant. Highly recommend.

Debbie's got the right idea - staying in the CBD is a great choice for proximity to the Quarter, without any of the baggage of actually staying in the Quarter. The only place she left out is an old favorite of mine, International House. It's modern and a little quirky, with a wide open lobby and a pretty fantastic bar. The rooms are quiet and affordable, if a little on the small side. The International House's sister hotel, Loft 523, is around the corner. The staff at both are great, and the nearby restaurants like Lüke and August are great, accesssible options when staying in the CBD.

The Hotel Modern: They have a great bar and a great place restaurant. It's right off the street car line, about a mile to the quarter- so walkable.

The Whitney Wyndham Hotel is a refurbished bank. The rooms are spacious if basic, but the restaurant, built into the former bank lobby, is nice. Free WiFi, centrally located in downtown.