* Protesters urge Catholic Church to give women more say
* Some cardinals in conclave want new role for women
* Some Catholic women happy to accept subordinate role
By Catherine Hornby
ROME, March 12 (Reuters) - Protesters demanding a greater role for women in the Roman Catholic Church set off a pink smoke flare on a hill above the Vatican on Tuesday as the men-only conclave that will choose the next pope began.
Mimicking traditional smoke signals from the Sistine Chapel - white for a new pontiff and black for an inconclusive vote - the women also wore pink clothes and "Ordain Women" badges.
Some women argue that they already play an important role in the Church, teaching and caring for young Catholics and doing much of its missionary work, while others say exclusion from senior roles and the ban on women's ordination is out dated.
"The current old boys' club has left our Church reeling from scandal, abuse, sexism and oppression," said director of the Women's Ordination Conference, Erin Saiz Hanna, one of a small group assembled on the Janiculum hill overlooking St. Peter's.
"The people of the Church are desperate for a leader who will be open to dialogue and embrace the gifts of women's wisdom in every level of Church governance," she said.
The Vatican says women cannot be ordained priests because Jesus Christ willingly chose only men as his apostles. Advocates of a female priesthood say Jesus was merely conforming to the customs of his times.
Tuesday's protest in Rome followed a pink smoke rally in New Orleans over the weekend, with similar events planned in cities across the United States in coming days.
Last year, Pope Benedict restated the Church's ban on women priests and said he would not tolerate disobedience by clerics on fundamental teachings. Under his leadership, the Vatican cracked down on advocates of female ordination.
But some cardinals attending the conclave this week have spoken out about the need to review the role of women in the Church and the leadership positions open to them.
Argentine Cardinal Leonardo Sandri, 69, told Reuters this month that women must have a much more important role in the life of the Church and be able to contribute in areas which are now only open to men.
At present women, most of them nuns, can only reach the position of under-secretary in Vatican departments, the No. 3 post after president and secretary.
Currently only two women are under-secretaries: Sister Nicoletta Vittoria Spezzati and lay woman Flaminia Giovanelli.
Spezzati holds the post at the Vatican's department for religious orders, which is run by Brazilian Cardinal Joao Braz de Aviz, who will vote in the conclave and is a potential compromise candidate for pope.
He has played a mediating role after the Vatican last year reprimanded American nuns for not doing enough to fight against abortion and gay marriage. He has also been credited with easing the heavy-handedness of his predecessor at the department who had complained about liberalising trends in the Church.
Giovanelli works for the Pontifical Council for Justice and Peace under Cardinal Peter Turkson from Ghana, the top African candidate for pope.
Some women who are tired of waiting for the rules to change have taken matters into their own hands. The Association of Roman Catholic Women Priests (ARCWP) say there are now more than 124 female priests and 10 bishops worldwide, though the Vatican considers them excommunicated.
Janice Sevre-Duszynska from the ARCWP, who attended the Rome protest in her white priestly robes, said if she met the next pope she would ask him for a follow-up to the modernising Second Vatican Council of 1962-65, which discussed relations between the Church and the modern world.
"I would say to him that we need a new Vatican council with no bishops being invited, no cardinals, no priests, but just getting the people from local parishes, and people who have come out of prison and homeless centres," she said. (Additional reporting by Ana Valderrama and Philip Pullella; Editing by Louise Ireland)