One person who has more or less been ruled out of contention is former deputy prime minister and New Zealand First leader Winston Peters.
Ardern said at Waitangi earlier this year that she was not keen on appointing a former MP to the job.
As well as ruling out Peters, this means a number of other would-be contenders are out of contention as well.
It means Christchurch mayor Lianne Dalziel, and former Labour MP, is out of the running.
As is the likes of former Labour MP Dame Winnie Laban, who is not the Assistant Vice-Chancellor (Pasifika) at Victoria University.
Despite this, she has been tipped by some as a favourite.
It also rules out current Speaker of the House Trevor Mallard – although he was never in serious contention, given his prominent role at present.
Dame Sian appears to be the front-running at this stage.
She was New Zealand's 12th Chief Justice – the most senior position in the country's judiciary.
She was the first woman to hold this position in New Zealand and before that, she and Lowell Goddard were made the first-ever women members of the Queen's Council.
She was made a Dame in 1999.
Before becoming a Chief Justice, she held a number of other senior legal positions and did a lot of work on Treaty claims and settlements.
Another more serious contender for Governor-General is Andrew Becroft.
Becroft also has a strong legal background, being appointed as a District Court Judge in 1996 and then becoming the Principal Youth Court Judge of New Zealand in 2001.
Appointed to lead the Children' Commission in 2016, he has since become a very public figure.
Another name being floated is Joe Williams – but he is not seen as likely, given he was just appointed to the Supreme Court of New Zealand.
Another person who is being talked about is Ella Henry.
She was a negotiator for her Iwi, Ngātikahu ki Whangaroa, and has been involved with the Post-Settlement Governance Entity, as Trustee and Chair, until her appointment as Director of Māori Advancement at AUT.
She has a PhD in Māori Entrepreneurship and a number of other qualifications.