Cardinal Brady also met with representatives from the Survivors of Child Abuse group.
Cardinal Brady said Fr Peyton had "a great gift" for communication and would have been "big into texting and Twitter" if that had been available in his time.
She and other members of her group met Cardinal Brady earlier this month to seek his cooperation in a forthcoming inquiry, ordered by the NI Executive, into allegations of clerical abuse in Northern Ireland.
When he was first unveiled as Cardinal Brady's likely successor, Monsignor Martin spoke about the influence of his large family, and the new cross and ring that he will wear as coadjutor Archbishop also bears a strong family connection.
Cardinal Brady said in his Christmas message that he hoped "that everyone who believes that the right to life is fundamental will make their voice heard in a reasonable, but forthright way to their representatives, reminding them that the right to life is conferred on human beings not by the powerful ones of this world but by the Creator".
The four Catholic Archbishops of Ireland, including Cardinal Sean Brady, have criticised the decision.
Monsignor Dooley's comments came as he spoke in defence of Cardinal Sean Brady.
Cardinal Sean Brady urged everyone who believed in the right to life, to put their views in writing to politicians.
Twitter - the social messaging network - or text or email is a fine way to spread the gift of prayer, Cardinal Sean Brady said.
It went on to pay tribute to politicians and church leaders, including Cardinal Sean Brady for his "strong and moving plea to the congregation and wider community".
In Ireland Cardinal Sean Brady, the primate, has admitted that he was present in 1975 when two teenage boys were persuaded to sign oaths of silence about their abuse by Father Brendan Smyth.
The head of the Catholic Church, Cardinal Sean Brady, said he also welcomed the news agreement has been reached on the devolution of policing and justice powers to the Northern Ireland Assembly and on a range of other issues.
However, as cases like that of Cardinal Sean Brady in Ireland prove, many of those now in the positions of authority in the church are the very same who stood by, or worse still, colluded in the cover-up of abuse.
CNN: Why next pope must open up church and usher in Vatican III
On Monday, the leader of the Catholic Church in Ireland, Cardinal Archbishop Sean Brady, said the attack on a Constable Kerr was an attack on all of society.
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