I think that with Pope Francis being appointed as Pope, the Catholic Church could easily stay relevant. Already he has a great view on the world and seems to be promising for the Church. He is sympathetic and understanding of the times that we live in. He is unlike his predecessors and I think time will soon tell how well he will help the Church.
Of course the new Pope will help the Catholic Church to remain relevant. There nearly 1 billion members of the Catholic Church, and many more that are intrigued by the selection of the new Pope. Why would the church need to change their beliefs in order to be relevant? Pope Francis is from Latin America where nearly half of all Catholics are and where members seem to be the most active in their faith. For good or bad the Catholic Church will always be relevant.
If the reputation of the Church has done nothing to its relevancy, then a new Pope, good or bad, won't either. It will remain relevant to the lives of some, no matter the Church's stance on current issues. For others it plays no part, as it always has been. If anything keeps the Church relevant in a broader sense, it is the media.
Whomever is chosen as the new Pope will likely not help the Catholic Church stay relevant because history has proven, through the inflexibility of the Church's stance on issues, that the Church seems to pride itself on not being relevant, but stodgily clinging to old, sometimes nonsensical, mysoginistic, blind (when it comes to the pedophilia problem) ways.