For stays of up to three months your passport should be valid for the proposed duration of your stay; you do not need any additional period of validity on your passport beyond this.
UK Emergency Travel Documents
UK Emergency Travel Documents are accepted for entry, airside transit and exit from Cyprus.
British nationals don’t need a visa to enter Cyprus. If you intend to stay for longer than three months, you should check entry requirements with the Republic of Cyprus High Commission in London or the Republic of Cyprus Ministry of Foreign Affairs.
Travelling with Children
See this leaflet on travelling with children produced by the British High Commission in Nicosia.