The Queen, who turns 92 on Saturday, told the 53 Commonwealth leaders and dignitaries present at the opening that it was her "sincere wish" that they decide Prince Charles should lead the group "one day".
Commonwealth leaders will discuss the issue of succession at a retreat at Windsor today.
The role is not hereditary and there have reportedly been some suggestions that the role be rotated around member states.
Speaking on gender equality, Clark noted how few women leaders were present at the Chogm opening.
"It's dire".
"We really almost need a he-for-she campaign from the Commonwealth leaders to encourage women's leadership."
She said political parties had to recognise the importance of gender equality in parliaments and governments.
"They are often a block to progress. They are promoting the 'old boys' into the safe seats, on to the lists."
New Zealand had female prime ministers for more than half of the two decades and Kiwis now had an expectation that a woman would be PM about half of the time.
"Who knows who it will be after Jacinda, but they will be there, the women are there," Clark said.
