In the story in which Rivkah comes to marry Yitzchak, Avraham's servant (whom I'll call Eliezer) tells Rivkah's family about how he met Rivkah that day. Even though the Torah has just told us how it happened, it nevertheless gives us the entire text of Eliezer's explanation - telling us the exact same story, almost word-for-word, all over again.
There are a number of minor, incidental differences between the original story and Eliezer's retelling: changes in word choice and order, omission of irrelevant background information, and so on. Here are two differences that seem, in my mind, to be significant.
1) In the original story, Avraham tells Eliezer to go "to my land and heritage" to find Yitzchak a wife. In Eliezer's retelling, Avraham tells him to go "to my father's house and family".
2) In the original story, Eliezer gives Rivkah jewelry and then asks which family she belongs to. In his recounting, he mentions asking about her family, and only then mentions giving her jewelry.
These two difference apparently have the same cause. From Avraham's instructions, Eliezer did not conclude that Yitzchak's future wife had to be from any particular family. Avraham had only mentioned "my land and heritage" after all; literally, this criteria would be met by any woman in Haran. Indeed, upon arriving in Haran Eliezer made no effort to seek out the "Nachor family", which shouldn't have been hard, but rather went straight to the well to find a woman with good character traits.
But in the end, it "just so happened" that the woman was from Avraham's extended family. Eliezer must have realized that this was how God, and perhaps Avraham, had intended it the entire time. Thus he rephrased Avraham's statement to explicitly mention the woman's family.
Similarly, Eliezer had given Rivkah jewelry as soon as she had watered his camels. As far as he was concerned, the test was over and she had passed. All that was left was to convince her and her family that the marriage should take place, and the jewelry gift was part of that. But once she told him the family name, he realized there was more to the story. In the end, the destined wife was the woman who watered the camels AND who was from Avraham's family. In Eliezer's retelling, he does not give the jewelry until learning that both these conditions are fulfilled.
We need not say that Eliezer intentionally lied in either instance. On one hand, given his excitement at the miracle which had just occurred, he could have stumbled in his speech or erroneously remembered what the plan had been before the excitement began. On the other hand, Avraham's instructions might have alluded to the family (why should Eliezer look for a wife in Haran, but nowhere else, if the reason is the Canaanites' evil?). And regarding the jewelry giving, we should remember that Biblical narratives are often arranged partly thematically and partly chronologically, so Eliezer's speech could simply mean that he now saw the jewelry giving as a consequence of Rivkah's family identity.
Now, a third difference between the accounts which I think matters.
3) In the original story, Avraham release Eliezer from his oath in the case that "the woman does not desire to follow you". In Eliezer's retelling, the case is rather that "they [the family] do not give [the woman] to you".
It seems that Eliezer is just being polite here. He is discussing Rivkah's future with her family. Since they seem to be in charge of deciding whom she'll marry, Eliezer credits them with the ability to prevent her from going.
And yet, in the end Rivkah *is* asked whether she wants the marriage. Thus Avraham's statement, implying that the decision will depend on her, turns out to be correct. This is similar to Avraham's hint that the wife will come from his own family, which also turns out to be correct.
It's not obvious from a superficial reading, but in addition to his other qualities, Avraham appears to have been a pretty impressive prophet!