Right and left mean north and south. And Ishmael lived south of Lot, while the plain was north of the Negev, where Abraham was.

A good direction for the question, in the name of my nephew, the brilliant young man Yekutiel Yehudah son of Rabbi David son of Rabbi Yonah son of Rabbi Yekutiel Yehudah, of blessed memory.

Eliezer is the focal point.
He was sent on a difficult mission: to persuade some family to make a match and send their daughter with someone who isn’t here, and who claims that he exists at a huge distance away, with no transportation or communication (thousands of years ago), by a route from which in reality there was then no way back.
And also to finalize the match without unnecessary financial costs (certainly not some absurd amount like 400 silver shekels, as at the beginning of the Torah portion). He gave “precious gifts” to her brother and mother, and that was it. (No virgin bride-price… a match without money…)

For this mission he uses two tactics.
1. He bluffs them that they have to give their daughter because Abraham specifically commanded that it be from his family. (The command says only “from my land and my birthplace”; it does not specifically say “from my family.”) And there are no other alternatives among the neighbors.
2. So they won’t think they can extort money for the match, because at worst there’s an alternative from Ishmael and from Lot.

Now he is standing in Bethuel’s house in the far north, in the region of Carchemish, and declares: If you give her to me (and she will have wealth and honor as the wife of…), good; and if not, I will turn to the right or to the left—Eliezer’s right and left. His right (roughly, without precise maps) is toward the southwest, toward Egypt, to Ishmael. His left is toward the southeast, toward Lot.
And now it’s understood what “I will turn to the right or to the left” means. True, I supposedly must specifically take from Abraham’s family (maybe a bluff), but I have an alternative from Ishmael and from Lot, to the right and to the left.

Bottom line: it worked, and they gave their daughter without financial compensation.

And regarding Rashi, who says “to Abraham’s right and left”?
It should be said that perhaps the original version was “to his left,” with the abbreviation standing for something else, and the copyists mistakenly deciphered the abbreviation as “of Abraham” instead of “of Eliezer,” as seems clear from the plain sense.

Now everything falls nicely into place.

Did Rabbi Michi write the first answer? That’s a really embarrassing answer.
First, because anyone who has studied even a little Hebrew Bible knows that “right” is a synonym for south. Nowadays maps are oriented with north at the top. In those times the orientation was eastward, which was called “forward,” apparently because a person is imagined as standing facing the rising sun. That is why the Mediterranean is called in the Torah the “rear sea,” meaning the sea behind.
Second, because it shows a dismissive attitude toward interpreting a verse in the Torah. One can say one doesn’t know, one can try to work it out, but this answer expresses contempt for the very question, as if saying: what difference does it make? who cares? Such an attitude toward understanding a verse in the Torah says mainly something about the person who wrote it.