I use hotels.com to book my rooms because I get a free room after 10 nights. I've already been able to get 4 free rooms.
I like to come back to Manhattan for theater trips 4 to 5 times a year, typically for two to three nights at a time. Times Square hotels can't be beat because there is nothing quite like running around all day, coming back to the room, showering, resting, and walking out of your room within 15 minutes of curtain.
Here are my favorites, based on the OP's categories, although I realize one person's economy may be another's "hell no!" and one person's luxury may be another's economy. So this is purely subjective, but I hope reasonable.
Luxury:
Westin Times Square
Moderate:
Paramount
Row (the HVAC units are too noisy for sleep)
Budget/Economy:
Edison
Mayfair
So bad I would never stay again (however they're cheap:
Manhattan Broadway Budget on W 38th
Hotel Carter