A word/phrase meaning the "house where I was born"?

Birthplace: was used to indicate the place where someone was born especially in the past. Nowadays is it more common to be born in the birth-centre of an hospital and birthplace usually refers to the village/town where someone is born.

Birthplace:

• The place where someone is born.

John Quincy Adams Birthplace:

• The John Quincy Adams Birthplace is a historic house at 141 Franklin Street in Quincy, Massachusetts. It is the saltbox home in which the sixth United States President, John Quincy Adams, was born in 1767 to Abigail Adams and John Adams.

A common expression to indicate your home is the house where you grew up referring specifically to the first years or the first part of your life.

Source: www.wikipedia.org

You can call the home where you were born and lived as a child your childhood home. This doesn't preclude being born in a hospital. Nowadays, newborns stay only one or two days in a hospital before being taken home by their parents

Johnny Cash's childhood home opens to the public

Country music legend Johnny Cash's childhood home has been opened to the public as part of a drive to revitalise the Arkansas town where he grew up.

The Cash family moved to the house in Dyess in 1935, when Johnny was three, as part of a government drive to help families after the Great Depression.

source: http://www.bbc.com/news/entertainment-arts-28827883

EDIT
If the childhood home is not the same home where one was born then the OP needs to specify more clearly. I would suggest the following phrase:

The home where I was born