May said training will be provided for young people taking part in NCS programmes to help them cope with exam stress, struggles at home and other challenges by raising their awareness of mental health and improving their knowledge of where to get help.
"Mental health issues can have a devastating effect on young lives and that's why making sure young people are fully supported both inside and outside of the classroom is a key priority for me," she said.
"We know that early intervention, along with giving young people the confidence to access support, is key - that's where NCS plays such a vital role."
In tandem, the NCS will train more than 10,000 adults who work on its behalf on mental health issues. Frontline workers will likely undertake an e-learning course on basic mental health awareness, while NCS programme managers will take one- or two-day course.
Work to develop the training programmes has not yet been commissioned and no date has been set for when they will be introduced. The NCS will also create a network of young graduates from its programme to champion mental health issues.
Michael Lynas, chief executive of the NCS, said: "As our country's flagship programme for 16-year-olds, we know just how important the issue of mental health is to this age group, and we hope this initiative will help the next generation to live healthier and happier lives."
More than 100,000 young people are expected to take part in NCS programmes this year.
May's announcement follows the introduction in June of training for secondary school teachers and staff on how to identify and respond to the early signs of mental ill-health among pupils.
The training for secondary school staff is due to be rolled out to all schools in England by 2020 and to all primary schools by 2022.
According to the Prime Minister's Office, more than half of all mental health problems start by the age of 14 and three-quarters by the age of 18.