I have used Stevens for many years, also for numerous visitors. They don't have a booth at the airport, but they will meet you just after Customs.
http://www.stevenscarrentals.com/
They are MUCH cheaper than anything from any of the booths and they are reliable. On one occasion, my daughter had a car that broke down, in a village 20 km from Larnaca. She phoned them and they had a replacement car there within about half-an-hour. Can't expect better than that.