www.easyhotel.com has just opened a branch in Dubai and if you book enough in advance you would pay around $40 a night - this is the basic price and they charge you for everything - even the hire of the remote control for the TV for the day! The location may not suit as its in JAFZA - Jebel Ali Free Zone - which is at the Abu Dhabi end of Dubai so to go anywhere, you will have to get a taxi or the metro.
I stayed at the Rose Garden Hotel Apartments in Barsha a few months ago. Nice apartments with kitchens etc and a good location - within walking distance of Mall of the Emirates which also has a metro stop. Downsides are it is a family run place so they stipulate no alcohol and no unmarried couples - we were asked to produce our marriage license!
Barsha is one of the new up and coming areas and there is certainly plenty of choice. Centro Barsha, the Holiday Inn, Seven Sands - all pretty much of a muchness and the earlier you book, obviously the better the rate.
I have had friends who stayed at Minc Apartments and rated it.
Depends whether you want to be near the airport or more into town. Deira (near the airport) has a lot of apart-hotels, but I have never stayed there.
Sheikh Zayed Road has some older hotels such as the Crowne Plaza, Ibis and Novotel which you can sometimes get good rates at. about 9 years ago