Lee, who died last year, is revered by Singaporeans, while his children are respected members of the establishment. His son, Lee Hsien Loong, is prime minister.
The family had presented a united front, but that cracked in recent days when Lee Wei Ling, a prominent doctor and newspaper columnist, publicly objected to what she saw as excess in the one-year commemorations.
She also accused the government - led by her brother - of exploiting his legacy for its own gain, which Prime Minister Lee has rebutted.
The exchange has shocked Singaporeans - public disputes among the establishment in this tightly controlled city are extremely rare, let alone one involving the country's equivalent of the royal family or the Kennedys.
It has also fed into ongoing discussions on the tricky issue of how best to remember Lee Kuan Yew, a beloved idol who hated being one.
Lee's death on 23 March last year prompted a massive outpouring of grief among Singaporeans who queued for hours to pay their respects, surprising even government officials.
At least 100 events were organised for the one-year anniversary, ranging from solemn ceremonies and a candlelight vigil to tree-planting and kayaking events.
Wax statues of Lee - widely known as LKY - and his wife were put on public display with flowers laid at their feet, a schoolbook teaching Lee's values was launched, while some ardent fans online even claimed to have seen his face in the clouds.
Two days after the anniversary, amid unease in some quarters about the commemoration's intensity, Ms Lee posted a disapproving piece on Facebook arguing that her father would have objected.
"Any veneration could have the opposite effect and lead future generations of Singaporeans to think that my father's actions were motivated by his desire for fame, or creation of a dynasty. He strove hard and determinedly in life to advance Singapore, and not for his place in history, or leaving a great legacy," she wrote.
She later accused the government of wanting to "glorify my father", and alleged that a newspaper was doing the government's bidding by refusing to publish the piece unedited. The paper, The Straits Times, strongly denied her claims of censorship.
In a now-deleted post of an exchange with the paper's editors, she wrote that her brother "has no qualms abusing his power to have a commemoration just one year after LKY died... if the power that be wants to establish a dynasty, LKY's daughter will not allow LKY's name to be sullied by a dishonourable son."
One Sunday, the prime minister  released a statement saying her accusations were "completely untrue", adding that his government felt the level of commemoration and activities "were generally appropriate".
In Singapore, Lee's admirers face the dilemma of how to honour a famously unsentimental figure who sternly disapproved of any worship - he refused statues and monuments in his name, and wanted his home to be eventually knocked down to prevent it from becoming a shrine.
In a 1984 interview, Lee had made clear his objections to the idea of a personality cult in Singapore, saying: "The danger... is that when the leader dies, you are faced with the difficult problem of finding another."
The Lee siblings' squabble has divided public opinions, with many taking to their Facebook pages to weigh in.
Some agreed that the tributes bordered on the excessive and were disrespectful.
"If he doesn't want to be hero worshipped then we shouldn't no matter what Singaporeans want. LKY has already given a lot to us. Let him rest in peace," said commenter Mohammad Nizam Abdul Kadir.
"The best way we can remember him is to continue as a united, multi-racial, progressive country. Uphold his values and principles," said another called Mano Sabnani.
But others argued that the round of commemoration activities was necessary for citizens, many of whom still feel a personal connection to Lee.
"I am one of the many who wanted to express my gratitude to him and also to remember him for his commitment and contributions to Singapore. I would humbly ask Ms Lee Wei Ling to at least indulge us as we miss him too," said Daniel Ng.
Carol Sim said: "Ms Lee has failed to understand that LKY does not belong to her alone. He belongs to the people of Singapore.
We respect your choice to remember your father in your own quiet way but you certainly cannot stop the people from wanting to do more."
A rare spat among Singapore's first family has left citizens agog and fuelled debate on how the city-state's founding father Lee Kuan Yew should be remembered.