Her stocks have risen since the demotion in 2014 of another potential National leader, Judith Collins.
That's a sensitive subject. Ahead of my meeting with the minister, her press secretary calls and gently suggests some parameters for an interview. The minister doesn't want to talk about any "boring" leadership questions. She would also prefer to talk about her portfolios rather than her personal life.
For someone who appears larger-than-life, Bennett can be nervy and self-conscious in person. "I don't want to talk to you," she says when I arrive at her office on the sixth floor of the Beehive. "I hear you've been talking to other people about me."
Colleagues, former colleagues and ministers I spoke to mostly sing her praises. She is a brilliant communicator and an unbelievably driven person, they say. Some, however, remark that she is a loner within the National caucus. She has no court or inner circle, which could be a crucial weakness to a potential leadership bid in future.
"Paula is not really close to anyone," one National MP says. "She is friendly with everyone ... but it's also hard to break through and get to know the real Paula."
There is even some resentment among some colleagues that she has not stayed in touch and mentored junior MPs as she has risen up the ranks.
Bennett says she does not hang around for after-work drinks.
"I don't socialise around here a lot. I spend the hours reading and working."
Her affability and humour can mask the fact that she is a hugely ambitious, serious politician.
"Sometimes my bubbling personality ... " she pauses. "People underestimate me all the time. It can be helpful sometimes. Because while they are standing back and laughing and thinking you're insignificant, you can just work a damn sight harder than them."
Parts of her personal background are already well-known. She is the solo mother from Taupo whose experience of welfare eventually drove her to become the minister of welfare.
Nothing about her early life marked her out for excellence. Her father Bob ran the village store by the lake in Kinloch and her mother Lee was a librarian. She was a stroppy teen who resented the way girls were treated compared to boys.
When she was given a sewing machine for her 16th birthday - her two brothers got shotguns - she traded the machine for a motorcycle. She left home the same year and went on the domestic purposes benefit.
Aged 17, she had her first child, Ana, who was later a young mum on welfare with a partner in jail. Two years later, Bennett bought her first home in Taupo for $56,000 with the help of a Housing Corporation grant and had to go back to work to cover the mortgage. She worked through the night as a waitress at a truck stop, in a hair salon, booking tours on Lake Taupo, and as a cleaner.
She pinpoints moments in her early 20s that caused her to turn her life around. At 23 she met Alan Philps, who would become her husband 20 years later in a very private wedding ceremony on Piha Beach - she wouldn't even confirm the groom's last name to media at the time.
Philps, a truck driver, "built me up and gave me a whole of confidence" at a low point in her life, she says.
"He's a neat bloke and grew up in a family with three strong sisters and a mother, so he had this real girl-power thing going on. He's your good classic bogan Westie, and behind that is this deep-thinking, almost feminist guy."
In 1991, she lost her brother Mark in a diving accident on an oil rig in Indonesia, her best friend was killed in a motorcycle accident, she broke up with her long-term boyfriend, and was fired from her job as a receptionist at a hair salon.
"That has you just completely re-assessing where you are at," she says. "It was feeling like my future was pretty inevitable. Which is what I think a whole lot of solo mums feel. It's already written, and changing that is just incredibly hard."
Bennett moved to Auckland to make a fresh start, cleaning dishes and later becoming a nurse aide at Beechworth rest home in Albany.
"It was the first time I was around a lot of very professional, very successful women. So I found them to be really empowering."
She was inspired to head to university where, after getting a taste of student politics, she switched from social work to social policy. "I thought, 'someone's got to make bigger change than that. Someone's got to change the system'."
She warmed to the National Party because, she says, its members did not judge her for her background or doubt she could go far. A highly-organised, energetic campaigner, she won the Labour seat of Waitakere at the second attempt in 2008. After the election, she "squealed" with happiness when she received a call from Prime Minister John Key telling her she would be social development minister. It was her "dream job" and the reason she came to Parliament.
So it marked a huge change when she left the welfare role in 2014 after six years. She later realised the daily encounters with child abuse and neglect had taken its toll. "One little girl I met who was 7 years old and cute as a button and she literally looks at you with these big eyes and starts telling you that bed's not a safe place. That sort of thing is pretty distressing."
Bennett did not ask to be climate change minister, and admits she knew little about the subject when she was appointed by Key in December.
The portfolio is a bit of a hospital pass. She is tasked with reducing New Zealand's emissions but because of political and economic sensitivities she cannot crack down on the biggest emitters - farmers.
"Two days in, I totally got why the PM gave me this job," she says. "I realised really quickly that it was as much about our identity and our country ... and who we wanted the world to see us as. Reducing emissions is an important part of that."
Green Party co-leader James Shaw says Bennett is handling the portfolio "better than anticipated" given she has no background in climate change. She ratified the Paris Agreement earlier than expected, has promised to scrap a government subsidy for climate polluters, and approached opposition parties to gain a cross-party consensus on some climate issues.
But the Herald's former economics editor, Brian Fallow, says Bennett has failed to show any urgency in her new job, despite growing international moves on tackling climate change.
"She says all the right things but all we've seen so far is [the removal of the business subsidy] and waffle."
Unlike climate change, Bennett asked for the newly-created position of Social Housing Minister because she saw it as a natural extension of her welfare role. She took over as the Government was struggling to make progress with its plans to sell thousands of state houses to NGOs.
She is faced with finding houses for the poor and homeless during a chronic housing shortage in Auckland.
"It's the first time where there's a big aspect that I literally just couldn't have control of. And that's property prices and supply," she says.
She wanted time to grasp the housing situation and craft a response but was forced to act more quickly. She won $41 million to pay for emergency housing in this year's Budget - the first time Government had funded the sector. Under pressure to come up with further solutions, she tentatively announced a new initiative to extend a $5000 grant to homeless people find housing outside of Auckland. When asked about it by reporters, Finance Minister Bill English was caught off guard. National MPs noticed her mistake. One said her handling of social housing had damaged any leadership prospects. Others disagree.
When I finally get to the taboo topic of her leadership ambitions, she deflects, saying she is just "grabbing it all and going with it". But it is undeniable she is ticking all the boxes for a possible promotion in future.
She says the secret to her rise to minister is to always have a back-up plan. Hers is to return to recruitment consulting, which she did immediately before becoming an MP. But you suspect that when Key finally stands down, a job in recruitment will be the last thing on her mind.
