That's a trick question, because a Dalai Lama is not chosen, he is found.
The Fourteenth and current Dalai Lama is a reincarnation of the previous Dalai Lama and was born Lhamo Thondup on July 6, 1935 in a small farming village in the Amdo region of northeastern Tibet two years after his last incarnation ended.
It is said that there were many signs pointing to Lhamo Thondup as the reincarnation of the Thirteenth Dalai Lama, including a vision a senior lama had pointing to the Amdo region and a small house with unusual roof gutters.
When senior lamas searched through the Amdo region they saw the monastery at Kumbum and then the small house with unusual gutters. When they entered the house they did not want to give away the purpose of their visit, but Lhamo Thondup recognized the senior lama of the delegation.
As a test, the lamas placed objects in front of the child that had belonged to the Thirteenth Dalai Lama and some that had not, and told him to pick up the objects he wanted.
As the story goes, Lhamo Thondup chose all of the objects that had belonged to the Thirteenth Dalai Lama. He subsequently passed all other tests and was determined to be the Fourteenth Dalai Lama.