The first inquiry chief, retired judge Elizabeth Butler-Sloss, quit after less than a week following questions about how her brother handled allegations of abuse by politicians when he was attorney general in the 1980s.
Her replacement, lawyer Fiona Woolf, stepped down in similar circumstances in October following complaints from victims' groups.
The departure of a second head had been a major embarrassment for the Government.
Ms May said there would be "no stone unturned" in the probe.
"I am now more determined than ever to expose the people behind these despicable crimes and the people in institutions that knew about abuse but didn't act, that failed to help when it was their duty, sometimes their very purpose, to do so."
Justice Goddard previously conducted an inquiry into the police handling of child abuse in New Zealand.
"I am honoured to be asked to lead this crucial inquiry and am well aware of the scale of the undertaking. The inquiry will be long, challenging and complex," she said.
"The many, many survivors of child sexual abuse, committed over decades, deserve a robust and thorough investigation of the appalling crimes perpetrated upon them."
A graduate of the University of Auckland, Justice Goddard was admitted to the bar in 1975.
According to website Professionelle, she was the first woman of Maori descent to be appointed to the High Court bench, in December 1995.
She is a member of the United Nations' subcommittee on the prevention of torture and from 2007 to 2012 she served as chair of the Independent Police Conduct Authority.
Justice Lowell Goddard
• Justice Lowell Goddard graduated LLB from the University of Auckland in 1974 and was admitted to the bar in 1975.
• In 1977 she commenced practice as a barrister and was appointed Queen's Counsel in 1988 alongside current Chief Justice Dame Sian Elias.
• In 1992 Justice Goddard was appointed Deputy Solicitor-General for New Zealand and also held the Crown Solicitor's warrant for Nelson District.
• She was appointed to the High Court bench in December 1995.
• From 2007 to 2012 she served as chairwoman of the Independent Police Conduct Authority.
• Justice Goddard is currently based at the High Court in Wellington.
• She has three stepchildren and one daughter, and is married to Christopher John Hodson QC.
- Source: Courts of New Zealand and Professionelle
- additional reporting AFP