Car rental in Dubai

It depends how much comfortable you are with driving. A confident driver would find his way as all major roads are well mapped. But if you are new to Dubai then may be hiring a car would not be such a good idea as driving here is very fast and driving standards are not up to the mark. Roads are very very busy during rush hours. Taxis are relatively cheap; readily available and drivers know their way around.

When are you planning to visit Dubai? As in Ramadan special discounts and promotions are made by different companies. If you are looking for cheap accommodation then try hotel apartments rather than hotels as they offer same services just at reduced rates. You can enjoy your privacy and extra space and all that at best locations in Dubai.
You can check My Dubai Stay. They have reduced their rates for Ramadan and also provide pick & drop facility from airport among other offers.
Good luck!

I would agree with most of what Sabeen says above (although she does sound a wee bit toutish), except the comment about taxis being readily available.

Availability of taxis is totally dependant on where you will be staying, and if, like most people, you want to be in the the centre of either Bur Dubai or Deira, then taxis are like gold dust, and frankly at peak hours you'd be more likely to find a leprechaun giving away maps to the end of the rainbow.

Car parking can be hard to find, but its much easier to find than taxis so if you're happy to drive then I'd recommend keeping the car while you're in Dubai.

As sabeen said, driving in the UAE can be pretty nail biting at times, but when I was out there (admittedly a few years ago now) the cabs were some of the scariest things on the road! I'd say keep hold of your rental car, but try to avoid the roads in town at the busy times, at least until you've acclimatised a bit. ;-)