Members states will be encouraged to put forward women candidates, after eight men in the role.
Former New Zealand Prime Minister Helen Clark is almost always mentioned in news reports as a potential candidate to replace Ban Ki-moon when his second and final five-year term ends at the end of 2016.
Prime Minister John Key said he had not spoken to Ms Clark about the job recently but if she put her hand up, New Zealand would back her "100 per cent". He thought she was considering her options.
She is the administrator of the United Nations Development Programme, No 3 at the UN, and has consistently refused to comment on the prospect of going for the job - no one has publicly declared yet.
The current process is very much behind closed doors. The General Assembly appoints the Secretary-General on the recommendation of the Security Council - the UN's highest body and responsible for world peace and security - on which Russia, China, Britain, France and the United States have veto powers.
It has traditionally nominated one person and the job has been on a regional rotation basis. If that continued, it would be Eastern Europe's "turn" although the prospects of finding a candidate acceptable to all, including Russia, are uncertain.
Agitation for a more transparent process has come from several member groups in the UN and non-Government organisations. Last week's resolution was worked up by the Working Group on Revitalisation of the General Assembly.
Changed process
• Next Secretary-General term begins 2017.
• Selection to take place 2016.
• Governments will be invited to nominate women.
• Candidates will conduct informal meetings with General Assembly.
• Security Council will make one recommendation, as now.
• General Assembly to approve, as now. Issues for next S-G: poverty reduction/climate change goals.
