"Twenty or 30 years ago was the age of revolutions and coup attempts in the world, but now it is the age of progress," says Zahra Eshraghi, one of the grandchildren, who believes it is not a good time for Iran to be isolated.
Ms Eshraghi is the most outspoken
family member but it is Hassan Khomeini, son of Ahmad, who many think could have a promising political future.
Said to have inherited
his grandfather's political ambition, the 37-year-old mid-ranking cleric has expressed frustration with some policies of a regime dominated by fundamentalists such as Mahmoud Ahmadi-Nejad, the president.
"We have to find new answers for new questions," Hassan said recently, reminding politicians that sticking to old rules is not always the best way.
As his grandfather did, Hassan has been studying in the holy city of Qom, and is expected soon to publish his first book on Islamic sects - a move that will lend him greater clerical authority.
He also runs his father's shrine and has, according to some analysts, resisted pressure to run for office
, including calls to contest the presidential elections in June. But he recently attended a gathering of reformist politicians, indirectly giving his support to potential poll contenders.
Radicals in the Islamic regime are watching him closely, and some lashed out at him
last year after he criticised the involvement in politics of the Revolutionary Guard, the elite force created by his grandfather as a counterbalance
to the regular army. Under Mr Ahmadi-Nejad, the Guard has expanded its political and economic influence.
In an extraordinary attack on a Khomeini family member, some radicals accused him of corruption, claiming that he drove a BMW, backed
rich politicians and was indifferent to the suffering of the poor.
While it might be thought ironic that the Khomeini family is today most closely associated with the reformist camp, some high-profile reformist politicians were close to the leader during his life. They became isolated from power after his death.
The Khomeinis are also tied to reformists through marriage. Ms Eshraghi is the wife of Reza Khatami, a senior reformist and brother of Mohammad Khatami, the former president and leader of the reform movement, which has long sought
to promote dialogue with the west and bring Iran back into the international fold.
Ms Eshraghi was barred
- by the Guardian Council, an unelected body that wields considerable power
- from running in the 2004 parliamentary election. One of her brothers, Ali, withdrew his candidacy from the general election last year, after what his family said was a smear campaign
against him. But many young Iranians are looking to Hassan, a good-looking man with a sense of humour who has befriended
athletes, including football stars. "He is young, full of motivation and a hard worker and he is trying to understand the country's youth," says one former Iranian official.
The shadow of the ayatollah still looms large
over Iran. Children are taught and tested on his legacy
and annual celebrations held in schools. Yet most Iranians are too young to relate to the revolution or understand its historical context. Beset by
economic troubles, international isolation and with their freedom repressed, many question their parents' allegiance to the revolution.
For some members of the Khomeini family, Hassan carries the hope of reviving what they insist is the true legacy of their grandfather - a democratic system under Velayat-e Faqih, the rule of the Islamic jurist which the ayatollah implemented.
Ms Eshraghi says Hassan should be aiming even higher than the presidency, because of his formidable "capacities". Does she mean that he is a potential future supreme leader, the ultimate position of authority that was occupied by her grandfather and is now held by Ayatollah Ali Khamenei? She smiles, but does not answer.
She is in no doubt, however, that Hassan has the legitimacy
in the Shia Muslim world to play a prominent political role, should he decide to do so. "He is very intelligent and will have a great future."