Greene (15) finished 4sec behind the Chinese champion in a time of 9min 59.20sec.
It was a personal best time by 12sec.
It was an impressive performance by Greene, who was competing against older athletes in a competition open to 19-year-olds.
It followed her gold medal-winning performance in the same event in her age group at the Pacific Schools Games in Canberra and the New Zealand College Games in Hamilton last month.
"The Chinese runner took off with 200m to go and I tried to stay with her," Greene said.
Greene is coached by former Australian coach Jim Baird.
The New Zealand team of 163 athletes won 44 medals, including 12 gold medals, in Sydney.
The five-day festival gave athletes aged from 13 to 19 a chance to compete in an Olympic-style event.
Competitors from 23 countries took part.
"It is encouraging to see the sports with solid youth-development programmes coming through with medal-winning performances," New Zealand chef de mission Barry Larsen said.
"I am confident that many of these athletes will be seen in London [Olympic Games] in 2012."The next Australian Youth Olympic Festival will be held in 2013, also in Sydney.
Singapore will host the inaugural Youth Olympic Games next year.
The festival impressed Dunedin's Andy Adair, who was head coach of the swimming team.
"The spirit of competition and the unity of the shared passion seem to have woven its way into the experience of Australian Youth Olympic Festival, making it something to remember," he said.
The only Otago swimmer in the team was Katie Kenneally (17), who achieved personal best times in the four finals she contested.
Her best performance came in the 200m individual medley, in which she finished fourth in 2min 21.18sec.
It was a personal best time by 5.89sec.
Kenneally was seventh in the 100m (1min 13.34sec) and 200m (2min 41.96sec) breaststroke events.
She was also 12th in the 400m individual medley in 5min 07.10sec, a personal best time by 4.7sec.
Athlete Toby Flett (16) was fifth in the 100m in 11.66sec and featured in the 4x100m and 4x400m teams that finished sixth and fourth respectively.
Biddy Skerten was fifth in the 400m in a personal best 61.81sec, and in the long jump with 4.67m.
She was also in the sprint and 4x400m relay teams that finished fifth.
Samara Gallaher and Caroline Crowley were members of the women's basketball team which lost the play-off for the bronze medal to China 95-64.
Kayaker Calum Brash was seventh in his second qualifying round of the K1.
Fact File
Otago medallists
Rebekah Greene (athletics), silver medal, 3000m (9min 59.20sec).
Caroline Kelly (cycling), bronze medal, 7.5km scratch race.
Caroline Kelly (cycling), bronze medal, 7.5km scratch race.
