Bobbisox posts some useful replies but that one is complete rubbish.
EU law requires all EU countries to admit any person who can prove that they're a citizen of a member state, by producing a valid passport or national identity card. Accordingly, any UK passport holder can enter any EU country right up until the final second of the stated validity of their passport. Further, since EU citizens have the right to live in any member country, it wouldn't matter (as far as the Cypriot authorities are concerned) if a visitor's passport ran out during their stay, since there is no requirement upon the visitor to ever leave the country. Obviously, though, it's wisest to ensure that a passport is valid on the planned date of return travel, in order to be able to provide the airline with valid ID and for re-entering the UK.
Chris