We were there a few years ago in Paphos and hired a jeep for 2 weeks. There was plenty of hire companies, we chose a local one rather than the big brands and never had any problems, had full insurance etc..the guy actually filled the tank for us and told us we could bring it back empty and wouldn't charge us.
Driving is great as on same side of the road and the views from the road out to sea are amazing.
------------- It is a wise man who has something to say.
It is a fool who has to say something.
Worth getting a price from a UK based broker, we often use Holiday Autos or Auto Europe. They will usually book you with a local supplier as opposed to the big brands. By arranging at home at least you know it's booked and paid for before you travel.
Try to find a company that lets you drive on the whole of the island,there is a lot worth seeing both north and south. We couldn't,we rented from Sixt,which I think is just a broker.