Before Isaac's birth, whom did Abram name as the heir of his house?

Answer: Eliezer of Damascus

Abram complained that he was childless and that "the heir of my house is Eliezer of Damascus," a servant in his household.

“But Abram replied, “O Lord GOD, what can You give me, since I remain childless, and the heir of my house is Eliezer of Damascus?” Abram continued, “Behold, You have given me no offspring, so a servant in my household will be my heir.””— Genesis 15:2-3 (BSB)