We have been visiting Cayman several times a year for 15 years and normally rent with Andy's. We have never had a problem with them. They are probably the cheapest if you intend taking insurance with the renter (their insurance rates tend to be lower than others). However, the cars are not the newest! There are many others to choose from (eg Avis, Budget, Economy, Coconut). They are all franchises.
In terms of cost, most rental company offer c10% internet discount off the rack rates. Note winter rates (mid Dec-mid Apr) are much higher than summer rates. If you rent 7 days you normally only pay for 6 (7th is free). On top of the rental rate, the company will try to sell you insurance (normally 3 levels are available) which can be expensive. On top of this, each driver will be charged for a local driving permit (c. US$7.5).
Most rental agencies have offices at the airport and at a location along 7 Mile Beach. They will pick up and drop off along the beach, but I'm not sure about at Georgetown.