Mothers' names are to be added to marriage registers in England and Wales for the first time as the government addresses "another inequality in marriage" by introducing the first reforms to the system in more than 150 years, David Cameron has announced.

In a wide-ranging speech on the family, in which he toned down his "broken Britain" rhetoric by saying that divorce can sometimes be the best outcome for children, the prime minister said the current marriage register system fails to reflect modern Britain.

Speaking at the Relationships Alliance in London, he said Britain had recently sent a powerful message by legalising equal marriage, but more progress was needed to modernise marriage.

"We're going to address another inequality in marriage too. The content of marriage registers in England and Wales has not changed since the beginning of Queen Victoria's reign. At the moment they require details of the couples' fathers but not their mothers. This clearly doesn't reflect modern Britain – and it's high time the system was updated. So I have asked the Home Office to look at how we can address this too."

The prime minister, who was accused of moralising during his time as opposition leader when he warned that dysfunctional families were helping to create a "broken Britain", adopted a more moderate tone as he said that he was not seeking to judge people's lives. He even said that divorce can sometimes be the best option for children.

Cameron said: "Let's be clear that there are some couples for whom splitting up is the right thing in the circumstances, however difficult the decision. In addition, there are also cases of domestic violence where what matters is making sure people are safe, rather than keeping a family together."

He went out of his way to praise "inspirational single parents" in another change in tone after he was criticised in opposition for appearing to belittle single parents by criticising the "couples' penalty". This was the Tory description of the top up to benefits to help single parents.

In his speech Cameron said: "When we talk about the importance of helping couples stay together – let's also be absolutely clear about the truly inspirational single parents in our country who do an amazing job bringing up their children."

But he also warmly praised the charity Relate, as he announced a doubling of the relationship counselling budget to £19.5m.

He said: "There are thousands of children who have renewed self-esteem and hope for the future because of your support in the toughest of times. And for families where breakups have been unavoidable, time and again your support has helped to make the fallout as painless as possible."

Cameron spoke of how his family lay at the heart of his life and his politics. He made a brief reference to his late son Ivan when he spoke of his luck in having four children. Ivan, who was severely disabled, died in 2009 at the age of six.

Cameron said: "For me, nothing matters more than family. It's at the centre of my life and the heart of my politics. As a husband and a father I know how incredibly lucky I am to have a wonderful wife and to have had four amazing children.

"But in loving my family, and in reflecting on my own upbringing, I've also learned something important about the way that family and politics are inextricably linked. Long before you get to the welfare state, it is family that is there to care for you when you are sick or when you fall on tough times.

"It's family that brings up children, teaches values, passes on knowledge, instils in us all the responsibility to be good citizens and to live in harmony with others. And so for someone from my political viewpoint, who believes in building a stronger society from the bottom up, there is no better place to start than with family."

But he joked that he sometimes failed as a father. "If people say: 'Hold on I saw you giving your daughter too much Coca-Cola' – guilty, we all make mistakes, we are all human. I am quite good on that one actually."

Eric Pickles, the communities secretary, will on Tuesday unveil plans to extend the government's "troubled families" programme to help vulnerable younger children from struggling homes. It will be extended to 500,000 "troubled families", costing the state more than £30bn a year – an increase on the 120,000 families that currently receive help.

• This article was amended on 21 August 2014. In an earlier version, the headline referred to "UK marriage registers". The first paragraph of the article itself was also amended to clarify that the changes apply only to England and Wales; mothers' names already appear on marriage registers in Scotland and Northern Ireland.