וַיֹּ֣אמֶר אַבְרָהָ֗ם אֶל־עַבְדּוֹ֙ זְקַ֣ן בֵּית֔וֹ הַמֹּשֵׁ֖ל בְּכָל־אֲשֶׁר־ל֑וֹ שִֽׂים־נָ֥א יָדְךָ֖ תַּ֥חַת יְרֵכִֽי׃
И сказал Авраам рабу своему, старейшине дома своего, который управлял всем, что у него было: 'Положи руку твою под бедро мое.
Rashi
זקן ביתו THE ELDER OF HIS HOUSE — Because the noun is in the construct state the ק is punctuated with a Patach.
Chizkuni
זקן ביתו, “the senior servant of his household;” Eliezer was old enough so that Avraham did not have to worry that he would take advantage of his position to indulge in incestuous conduct.
Ibn Ezra
Under my thigh. This was the custom of the time. It was a way of indicating that the underling was under the master’s authority and his hand was his to do with as he wished.