Andy's dealt with us in a devious matter (explained in detail on another thread). Fortunately, I pulled the plug on the transaction before signing anything and instead took the family across the street to the Avis counter at one of the major hotels (can't think of which one). Avis's cruise package was competitively priced and the employees were professional and provided excellent customer service. In fact, when we returned the car, it had to be returned to a different site. At that location there wasn't a driver available to take us back to the port, so the desk employee seeing that it was late in the afternoon and that he was not expecting any other customers, contacted his supervisor to ask him if it was okay to close the counter and drive us himself.
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.
I rented from Avis at the airport last year. If you call or email them, they will pick you up in Georgetown at the time of your choice. We waited at the designated pickup spot, and thought they had forgotten, but later determined that we were wrong about the time zone. So, they probably showed up afterall. There are numerous taxis in Georgetown, so we just grabbed one to the airport, very inexpensive. When we returned, they quickly shuttled us free back to Georgetown. I don't recall the rental price, but it was reasonable. There was a fee of $10 or so for a temporary driver's license. Be aware, it is left-hand driving there, but it is a small island with light traffic in most places. It does help to have a passenger making sure you don't slip up (easy to do, no matter how comfortable you are). i highly recommend a few hours on the beach at Rum Point on the north side of the island. Not much other place to go.