Mr McCully said he wanted to find issues on which there had been little movement and in which it could be possible to inject some momentum.
"A good example is the issue of Cyprus, which has been around for over 40 years."
The issue will come on to the Security Council agenda in January and again in July for a six-monthly renewal of a mandate for UN peacekeepers, which have been stationed there since 1964. At present it has 860 troops and 67 police.
"The question you have to ask is, if nothing has moved over 40 years, do we just wave this through again or do we say, hang on, let's just see if we can inject some new life into this?"
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About the council
•The Security Council is the UN body responsible for international peace and security.
The Security Council's first working day for the year begins on January 2, the first closed consultation meeting on January 5 and the public sitting could potentially be on January 6.
The council comprises five permanent members and 10 elected members and they take turns in alphabetical order to chair it for a month.
New Zealand will chair the council in July next year and again in September 2016, the latter including the week that leaders attend a UN session. That will provide an opportunity for Prime Minister John Key to chair a session or two that could include the leaders of the United States, China, Russia, Britain, and France.
US President Barack Obama chaired a session in September on foreign fighters.
Chile will be first up in the chair in January 2015.
Each country tries to pursue one or two thematic debates during its month chairing the council. Chad chaired it this month and besides dealing with other business, it led debates on the linkages between terrorism and international crime, and another on the council's relationship with the African Union.
One of the first things Mr McCully asked the Ministry of Foreign Affairs and Trade to do when New Zealand was elected in October was to begin thinking about themes the country could promote when it holds the chair.
"You need to decide in enough time to be able to signal your intention to other countries ... particularly if you expect them to contribute and especially if you want ministerial involvement in your initiative."