He has proposed many times in the past years, but this time, during a summertime stroll around the Hogwarts lake, she accepts his proposal. He purchases a cottage in Hogsmeade for them to live in so that Minerva can easily travel to work every day. McGonagall keeps her maiden name, much to the surprise of some in the Wizarding community.
According to Pottermore:
Shortly after Voldemort’s first defeat, Elphinstone, now white-haired, proposed again during a summertime stroll around the lake in the Hogwarts grounds. This time Minerva accepted. Elphinstone, now retired, was beside himself with joy, and purchased a small cottage in Hogsmeade for the pair of them, whence Minerva could travel easily to work every day.
Voldemort's first defeat was in October of 1981, so the next summer would be 1982.