O'Connor broke the British record as she finished behind Hungary's Katinka Hosszu in the 200m individual medley.
The relay team were second behind the USA for GB's sixth medal in Rio.
Michael Phelps swam the final leg for the Americans as he claimed his 21st Olympic title, having earlier won the 200m butterfly.
Away from the competition, a bus carrying journalists between venues had two windows smashed after stones were thrown at it, while it was confirmed Irish boxer Michael O'Reilly had failed a drugs test and will not fight at the Games.
It was another day of stunning action and incident in Rio:
Waiting game for GB - The late-night double medal haul ensured Britain did not draw a blank after canoeing contender David Florence finished 10th, the eventing and women's gymnastics teams finished out of the medals, and diving prospects fell short, too.
Tennis turn-up - World number one Serena Williams was knocked out of the women's singles by Ukrainian Elina Svitolina.
Not Sonny's day - Joint favourites New Zealand were beaten by Japan after Kiwi star Sonny Bill Williams was ruled out of the rugby sevens competition through injury. Britain just held off Japan for a second win as the men's event made its Olympic debut.
Pool mystery - Green, not gold, was the talking point in the diving where the water mysteriously changed colour, from azure to a swampy green.
O'Connor overcomes adversity for silver
London had already been awarded the 2012 Olympic Games by the time O'Connor started swimming at the age of 10.
Born and raised in Somerset, she has achieved spectacular success despite suffering from ulcerative colitis, a chronic bowel disease.
O'Connor is the first British woman to win an Olympic medal in the 200m medley and nearly caused a huge surprise as she closed in on Hosszu, who has now won three gold medals in Rio.
"It doesn't seem real but seeing my family, I was getting choked up," she told BBC Sport. "It means everything. It has been tough but it is so, so worth it."
Her proud father Sean held back tears as he paid tribute: "I don't think it's going to get better than this for me, to be honest. You're more proud of your children doing well than your own achievements."
The men's relay team of Stephen Milne, Duncan Scott, Dan Wallace and James Guy were second to a dominant USA.
Guy, who overhauled Japan's Takeshi Matsuda on the final leg, said: "The team is young and we'll be ready for the title at Tokyo 2020."
Analysis - BBC chief sports writer Tom Fordyce: "A second special night in the pool for Britain's swimmers. After Adam Peaty and Jazz Carlin two nights ago, this was another story of personal bests produced when it mattered most, of the best possible result being taken on the biggest stage of all.
"No-one was going to catch Katinka Hosszu, just as no-one was going to catch a US men's relay team featuring Ryan Lochte and Phelps.
"In taking silvers behind both, O'Connor and the quartet of Milne, Scott, Wallace and Guy have put British swimming wonderfully back on course."
Where does that leave Team GB?
Back in business after frustration earlier in the day.
Britain - with one gold, three silvers and two bronzes - are 11th in the medals table, which is topped by the United States, followed by China and Hungary.
