The situation has slightly
improved since the early years of partition and a few hire companies will allow their cars to be taken into North Cyprus
(not the larger multinationals, I think). I'm afraid you'll need to check with each individual company. There isn't one overarching policy and the rules of individual companies can and do change.
The primary drawback is that the additional insurance you must have (and will pay the company an additional fee for) will almost certainly be 3rd party only. Having 3rd party only cover (anywhere) is pretty risky imo...you'll be personally liable for any and all damage to the car (and the other person's car if there's a crash). If your insurance isn't acceptable you'll be stopped at the border.
There are potential workarounds. If you want to spend several days exploring the north you could drop off the southern-hired car in Nicosia
, use the pedestrian border crossing in the centre of that city and hire a car once you are in the north, with a new southern car hire starting on your return. Or, if it's only a visit of a day or so, you could perhaps just leave the southern car in Nicosia (maybe at a branch of the hire company?). It would be best to organise all that before you arrive.
Cyprus has a network of local and intercity buses so, depending on where your southern base is, you may be able to combine a bus trip to Nicosia with hiring a car in the North for a few days. Bus info here: http://www.cyprusbybus.com/
If you're spending the whole trip in the North it's easiest to just book a transfer from and to the airport (Larnaca
is easiest) and hire a car once you're there.
Note that cars hired in the North are, as far as I know, not allowed into the South.