The Taoiseach has told the Dáil that he thinks the name of the new children's hospital should be a simple one like 'The National Children's Hospital' or that it could be an Irish word.
Sinn Féin leader Gerry Adams suggested that the new hospital be named after Kathleen Lynn, in recognition of her role in delivering medical care to women and children and for her role in the 1916 Rising.
Leo Varadkar said he was not keen on the title being a political figure, despite suggestions that he had heard of both Ms Lynn and WT Cosgrave.
Labour leader Brendan Howlin said his party was suggesting a schools competition to name the hospital.
It was originally proposed that the hospital be known as the Phoenix Children's Health Ireland, however, this name clashes with an existing facility in the US.
The Taoiseach told the Dáil that his favourite name for a hospital was the Starship in New Zealand, which became the official name of the hospital there after children kept calling it the Starship.
Opened in 1991, Starship was New Zealand's first dedicated children's hospital, with its design chosen to appeal to children and young people.