Every person is Hashem's shaliach in the world

בקשת אליעזר הי' רק שלא יהי' הוא מעכב ח"ו בשליחת אדונו.

The request of Eliezer was only that he himself should not be an obstacle, chas ve-shalom, in his master's shlichus (mission).

When Eliezer davened for success in finding a wife for Yitzchak, his sole concern was that he personally not stand in the way of fulfilling Avraham's mission.

וכן צריך כל א' לידע כי הוא שליח בעוה"ז לעשות רצון בוראו.

And so too every person must know that he is a shaliach (messenger) in this world to do the ratzon (will) of his Creator.

Each person should recognize that he was placed in this world as Hashem's messenger, sent to carry out the ratzon Hashem.

וגם השבועה משביעין לכל אדם תהי צדיק ואל תהי רשע כו'.

And also the oath that is administered to every person: "Be a tzaddik and do not be a rasha," etc.

Chazal teach (Niddah 30b) that before birth every soul is made to swear "Be a tzaddik and be not a rasha" — the mission is sworn into us from the start.

ובאמת אין נגמר השליחות עד שפועל לצאת מכלל ארור לברוך ע"י האמונה וכבר כתבתי בזה במ"א באורך:

And in truth the shlichus is not completed until one accomplishes going out from the category of "cursed" into "blessed" by means of emunah; and I have already written of this elsewhere at length.

A person's mission is only fulfilled when, through emunah, he moves himself out of the realm of "arur" (cursed) and into "baruch" (blessed) — echoing Eliezer's own transition from Canaan's curse to becoming "baruch Hashem."

Summary: Eliezer's prayer was simply that he not obstruct his master's mission — a model for every person, who is a shaliach in this world sent to do the ratzon Hashem. Sworn from before birth to "be a tzaddik and not a rasha," a person completes his shlichus only when, through emunah, he lifts himself out of the category of "cursed" into "blessed."