In the Parasha of Hayyei Sara it says: "And Abraham was old, advanced in years, and G-d blessed Abraham with everything". Rabbenu Bahyei says that Abraham Abinu (the Patriarch Abraham), 'a"h, was blessed with much, including a long life. However, he had not yet been blessed with grandchildren, which is something he desired very much. He, therefore, made his servant Eli'ezer swear that he would find an appropriate wife for his son Isaac (Yis-haq Abinu, 'a"h).
The reason for requiring an oath from him was that Abraham Abinu was afraid that he might die before Eli'ezer accomplished his task. By making Eli'ezer swear, he felt that Eli'ezer would fulfill the task and he knew that his son Yis-haq would always listen to Eli'ezer and would, therefore, marry the girl.
The question is, why did Abraham Abinu not make Yis-haq Abinu swear instead? The answer is that he was concerned that if he did not place the servant Eli'ezer under an oath, he would find numerous excuses as to why he could not go get a suitable wife for Yis-haq Abinu. He made Eli'ezer swear, therefore, to ensure that he would take the responsibility seriously and find a wife for his son Yis-haq, who in turn would, as we mentioned, do Eliezer's bidding and would have no need to swear also.
(Rabbenu Bahyei on the Torah, Hayyei Sarah)
