According to Vatican Insider:

“In the election, I had seated next to me the Archbishop emeritus of Sao Paulo and the Prefect emeritus of the Congregation for Clergy, Cardinal Claudio Hummes, a great friend… When things became a bit precarious for me, he comforted me. And when the votes reached the 2/3 level, there was the expected applause.   He embraced me and said to me: ‘do not forget about the poor!’”

Francis continued: “While the voting continued, immediately I thought of St Francis of Assisi … Then I thought about war and so a name came to my heart: Francis of Assisi, a man of peace, a man of poverty, a man who loved and protected creation, and the poor. How I would love a Church that is poor and for the poor!”

The Pope said that once the election occurred, a cardinal suggested he take the name Adrian, because the last Pope with this name was Adrian VI, the reformer.  “Another told me: call yourself Clement XV, so that you can get back at Clement XIV who suppressed the Society of Jesus”. But in the end the choice was Francis.”