Bob, we holiday in Cyprus frequently. We go to a small fishing village called Pissouri. Beautiful unspoilt area in easy driving distance of the airport and the bigger towns and the Troodos mountains. The last two occasions we booked a 2 bed villa there, can't remember the name off the top of my head, but I'll look for the leaflet when I go home if you're interested.
We've just come back from Menorca (our 7th in 9 years), stayed in Cala'n Bosch in the SW. Had a villa, reasonably priced, ate out at the marina every night (~60 Euro for 4), but flights were eye watering. That apart I love menorca, quite quiet, but superb beaches and waterspouts, friendly locals and no groups of chavs.