Finding that budget for "rent" in a "good neighborhood" for a furnished place is going to be hard. It's going to take a lot of searching...
The "good" areas that I'm familiar with are:
East Oahu, Kahala, Kaimuki, Kailua, Manoa...perhaps.
Keep in mind everything here costs more... groceries too.
Perhaps, contact any churches.... and they might have people in the congregations that have "rentals" or might know someone with a house who might rent it out if they need money themselves. Housing here is in high demand, especially affordable housing.
Other than "youth hostels", rentals are high here, in good areas.
Also, you have kids, so you want someplace "safe" and with low chances of robberies etc.
Or, you can try and contact "Bed and Breakfast" places, and negotiate a "rate" for your family, since it would be for an extended stay and for the duration of the "winter season" and I'm sure given the economy with not many people traveling, they might just appreciate having the business... in tough economic times and in a "tourist" destination like this... businesses are perhaps more "open" to negotiating "deals." Know what I mean? It's worth a try...
Or try the Honolulu YMCA's or YWCA's here locally... they have "housing" but I don't know their "rules" or if they accommodate families. They do have websites too.
Good luck!