One born in my house. — This is a mistake. Those born in Abram’s house were his servants (Genesis 14:14). The Hebrew is, the son of my house, my house-son, not born of me, but the chief of the house next to myself, and its representative. Eliezer was probably born at Damascus.
Related Commentaries of Genesis 15:3
Genesis 15:3
3 And Abram said, Behold, to me thou hast given no seed: and, lo, one born in my house is mine heir.