As we rarely get day after day of awful weather it isnt really a big problem.
We tend to get a day or two of heavy rain with occassional sunny spells in bewteen then we get some nice weather again.
Mind you at the moment it is bucketing down, I dont remember such heavy rain for the past two or 3 years.
After the rain it is always so fresh and we like to go out for a drive and enjoy watching the countryside getting greener and the fabulous wildflowers springing up.
If you do want things to do when it is raining you will find most hotels have fitness centres that you can join, with indoor swimming pools, jacuzzis etc.
There is also a branch of the UKCA in Larnaca I believe and they will have all sorts of activities to keep you busy.
Do what you would do in UK, go to the Cinema, there are several that have films in English, or light the fire, open a bottle of wine and watch an old movie on the TV ( or DVD) As I write this we have rain pouring down for the second day, and more is promised.