"There's plenty of time to travel but houses are just going to get more expensive. It's a long term investment," Nelson said.
"If you've got money in an account or an investment, you can lose that investment or be tempted to spend that money. Whereas when it's in a house you can't spend the money because the money is in the house."
Nelson works two part-time jobs, on top of volunteering at her local hospice shop, to help save money to renovate and pay off the mortgage on the house.
Nelson and her partner also live in the house with a boarder, to help pay the bills.
She said the bad reputation millennials had for frittering away their money wasn't fair.
"It's the same with any generation. There have always been people who have not wanted to do work or not wanted to save…but there are a lot of people who are wanting to work hard and save and get places.
"A lot of millennials are realising that they need [to save] money as a backstop and they do want to save, it's just trying to be able to. It's not always possible with the fact that costs are going up a lot and there's not a lot of permanent work available."
The research also found many millennials plan to buy a home, but instead of saving for a deposit they are putting their money aside to travel overseas, go on holiday or buy a car.
"Millennials are motivated to save but just by different things. When you're in your late teens and early 20s you have a different set of financial priorities and plans than older people," said BNZ's director of retail and marketing Paul Carter.
"Buying your first car, going on your OE or travelling and seeing other parts of the world are things that most of us have done when we were the same age.
"It's great to see the millennials are saving for these things rather than slapping them on credit and getting into debt."
Carter said it has also become more expensive to buy a home, meaning people need to give it more thought and planning than they used to.
He said the trick to saving was to set up two pots - one for short-term goals like an overseas trip and the other for a future home or other investments.
Almost 70 per cent of young people also said they have the capacity to save some money and almost a third were motivated to make a budget and stick to it.
"These are all good signs, particularly for those who'd like to own a home one day," said Carter.