In the past 60 years the meeting of cardinals called conclave (with key) lasted very little, the longest was 5 days. This time it has a special ingredient that is repeated almost after 600 years: not elect a successor of a dead Pope but alive and declared one Emeritus, Benedict XVI, who resigned and in his final statement before his retirement, offered respect and obedience to the new Supreme Pontiff.
According to the known procedure, as soon as the older Cardinal make the announcement of “Habemmus Pope” and the name they have chosen, reiterating the tradition since the year 533, the new Pope would improvise a speech, thank and greet the faithful Catholics, and perhaps to give some insight of his line of conduct for the church of Christ.
The assumed name chosen is a personal matter and could answer, according to experts, to the desire to continue with an intellectual philosophy of any predecessor or, also, by devotion to a saint. The constitution does not forbid to the Catholic Pope to keep his name, but tradition says that he will take another.
Why not call himself Peter? According to experts no Pope has decided to call himself Petera because there is a respect for that name. Besides Peter also called himself Simon but Christ called him and said “Peter, upon you I will build my church.”