I believe the most popular understanding of the servant's action is that the request for water was a character test to select a wife for Yitzhak (see for example N. Leibowitz, 1976, pp. 223-229). The idea being that if Rivka offered to give water to the camels on her own then this showed that she was a kind and considerate person, and this would make her an excellent choice for a wife for Yitzhak. However, this approach is problematic.
First, one act of kindness is not enough proof that a person is a suitable wife. Maybe the test was a fluke. Maybe she was willing to provide water for the camels since she saw the wealth that the servant had brought with him. Would you marry somebody based on their answer to one question?
A second problem, as mentioned by the Radak (on 24:14, see also the Midrash Leviticus Rabbah 37:4), is that maybe the person who passed the test would be a slave and hence not a possible wife for Yitzhak. After Rivka gave the camels water, the servant asked whose daughter are you? Rivka told him that she was daughter of Betuel, 24:22,23 and from this information the servant would have learned that she was not a slave. Yet, this answer was insufficient since there could have been other social problems (married, engaged, going out seriously, etc.), that would have made her ineligible for Yitzhak that could not have been ascertained by just knowing who her father was. For example, 24:16 records, from the perspective of the "narrator" that Rivka was a virgin, as apparently this fact was important for the match, but the servant could not have learned this information from the test.
A third problem is that before the servant asked Rivka about her father, he gave her numerous presents, 24:22. Thus, before he knew that she was not a slave, he already had given her presents, which would have been forgone if she was a slave. In order to answer this problem, Ramban (on 24:22) explains that 24:22 only means that he took the presents out, but he did not give them until she told him that she was the daughter of Betuel. Ibn Ezra (on 24:14) writes that the verses are recorded out of chronological order, as according to him, the events described in 24:22, the giving of the presents, occurred after Rivka answered the servant's question, 24:23,24. These answers are difficult. The simple reading of the Torah is that the servant gave Rivka the presents before she told him who was her father.
A fourth problem is that it is not clear if Rivka's offer of water to the camels was such a generous offer. 24:32 records that water was brought to the men who accompanied the servant to enable them to wash their feet. While it seems obvious that the servant had men with him since he took ten camels (24:10), this is the first time these other men are mentioned. Were these men with the servant when he tested Rivka at the well? If yes, then it is odd that she offered water to the camels and not to the men. More likely, when the servant went to the well the other men stayed behind. The servant would then only have had a few (two?) camels with him when he made the test. If this is true, then Rivka did not have to give water to ten camels, which made the test easier for Rivka since she just had to offer and give water to a few camels.
Finally, there are many coincidences in the story. Of all the possible choices for a wife, it seems quite coincidental that the test would just happen to be done to Rivka, Yitzhak's cousin. Furthermore, 24:15 records that just when the servant finished stating the test, Rivka appeared. If she would have come to the well fifteen minutes later, would Yitzhak have married somebody else?
A second (and I believe the older) approach to understanding the actions of the servant is that this is a case of divination (see comments of Rav in Hullin 95b and Rambam, Mishnah Torah, Laws of Idol Worship, 11:4.) Divination is when a person acts based on some arbitrary sign, and here the sign would have been Rivka's statement in response to the servant's request for water. With this approach it makes sense why the servant gave Rivka the presents before finding out about her background since with divination one is relying on the sign, and no other information is necessary (see Kesef Mishnah on Rambam 11:4).
However, the divination approach is also difficult. 24:21 records that while Rivka was providing water to the camels, the servant wondered whether he would succeed in his mission of finding a wife. What was the doubt if she had already passed the test? Furthermore, by divination the sign must be exact, and Rivka did not exactly pass the test. The test was that she would say "Drink and I will waters the camels too," but she said drink, gave the servant water and only after the servant drank did she offer to water the camels, 24:18,19. She was supposed to immediately respond to the servant with an offer to give water to the camels, and not to wait until after the servant had drunk water. While this lapse might seem nitpicky, by the case of divination, due to the fact that the sign is arbitrary, one must pass the test exactly as stated, and when the servant repeated the incident he changed the order of events that Rivka offered to give the camels water prior to giving the servant water, 24:46.
Both of these approaches are based on the premise that the servant had no idea who Rivka was when he did the test, but the Torah gives many indications that he knew who she was along. One, 24:17 records that he ran to her. Why did he need to run to her and not ask the first girl who came to the well? Clearly, the servant wanted to do the test on Rivka. Two, as mentioned above, 24:22 records that the servant gave her the gifts before asking her about her family and this was because he knew who she was. Three, 24:23, records that when the servant asked Rivka who her father was, he did not wait for her answer, but he also asked if he could stay by her father's house for the night. We see that he desired to talk about the marriage even before she told him of her lineage, and this was because he knew who she was and her lineage. Four, the Torah never records that the servant asked Rivka her name or that he was told her name, but in 24:45, when recounting their meeting, the servant referred to her as Rivka, which again proves that he knew who Rivka was before the test.
I have been asked how could the servant have known who was Rivka? The answer is that the Torah records that he went to the well prior to meeting Rivka, 24:11. At the well he was able to learn who she was, just like Yaakov later learned who was Rahel when he came to the well, 29:4-6. In that case, the Torah specifies that the people at the well pointed out Rahel to Yaakov, while here this information is not recorded, but here there was no need for the Torah to mention this information since afterwards, as I mentioned above, the Torah makes it obvious that the servant knew that he was testing Rivka.
Rivka, Yitzhak's cousin, was always the destined wife since she was from the family of Terah, see 22:20-23 and our discussion on 24:4, "All in the family" (https://lobashamayim.blogspot.co.il/2009/11/bereshit-244-hayyei-sara-all-in-family.html). Avraham did not refer to Rivka when he commissioned his servant since if she would not have agreed to come to the land of Israel, then the servant could have chosen other members of the family. Even if one believes that the choice of a wife was really open to all, still a wife from Avraham's family would have been the preferred choice.
However, just knowing the desired wife did not make the mission simple since the servant had to get the family to agree to the match, and he had a problem that the prospective groom, Yitzhak, was not with him. Not only would the family not meet Yitzhak before the wedding, but due to the great distances between Nahor and the land of Israel, it is likely that they would never ever meet Yitzhak or see Rivka again. How could he get the prospective girl and her family to agree to the match without seeing Yitzhak? This apprehension is evident when the servant spoke to Avraham about the mission. He wondered what to do if the woman did not agree to come to the land of Israel, 24:5. The servant did not question whether he would be able to find a wife, but rather whether he would be able to convince her to come back to the land of Israel. Avraham told him that in such a case he would be exonerated from his oath, but it unlikely that the servant was completely reassured.
Accordingly, the servant devised a plan to get the family's permission for the marriage. His plan was that he would stage a test which would make it seem that her selection was destined from G-d. He was relying on the family's belief in divination, and in 30:27 we see that Lavan had this belief even after this incident. With this idea, we can understand the flow of the test.
When the servant reached Nahor, he stopped at the well, 24:11, where he learned who was Rivka. (Did he also learn the information recorded in 24:16?) The servant then prayed to G-d, both as a prayer for assistance and to set the test as a sign for G-d, and stated the test, 24:12-14. The servant timed his declaration of the test with Rivka's appearance at the well, and then just as he finished stating the test, he ran to her, 24:15,17.
The servant asked Rivka for water and she gave him water, and then she offered to give water to the camels, 24:17-20. As mentioned above, the delay in offering water to the camels was a small failure, but this was immaterial because the servant would be able to fix the small discrepancy when he would speak to the family. When Rivka was giving water to the camels, the servant wondered whether he would succeed in his mission (24:21) since the main part of his mission was not to find Rivka, but to convince her family to agree to the match, and this was the next stage of his mission. The servant was wondering whether G-d had decided to help him that his staged test would convince the family. (This reference to G-d might refer to his prayer in 24:11, but more likely refers to Avraham's declaration that G-d would help the servant in 24:7.)
The servant then gave Rivka the presents to get her approval to the match even before she told him that she was Betuel's daughter since he already knew this information, 24:22. He then asked here who was father and could he stay at her family's house, 24:23. Rivka answered that she was Betuel's daughter, she was a descendant of Nahor (the son of Terah), and that he could stay at their house, 24:24,25.
The servant then bowed to G-d, and said, "Blessed be G-d, the G-d of my lord Avraham…" 24:26,27. This prayer marks the completion of the first part of the mission which was to get Rivka's approval to the match. An almost identical phrase occurs in Ruth 2:20, when Naomi is thankful that Ruth had successfully been introduced to Boaz. In both cases, the crucial issue was the ensuing act, here the servant meeting Rivka's family, and there, when Ruth would go to Boaz in night. In both cases, the thanking of G-d was also a prayer for the successful fulfillment of the second and more difficult part of the mission.
Once the servant met the family, he immediately told them about the test without even eating first, 24:33. The servant began his re-counting of their meeting by mentioning that Avraham had been blessed, and that that Avraham had told him to pick a wife from his father's family, 24:34-38. Perry (2007, p. 302) wonders why the servant told Rivka's family this information since if the wife had to be from Avraham's father's family, there was no point to doing the test by the well. Did the servant slip-up? My guess is that this addition was to demonstrate to the family how Rivka was destined to be Yitzhak's wife. Not only was she Avraham's choice, but also when he did the test, which in theory was open to everybody, Rivka was chosen.
The servant then re-told all the narrative from 24:5-27, 24:39-59. The Torah records his recounting of the test at length since this was the crucial part of his mission. In his recounting of the test, the servant corrected Rivka's action as he said that Rivka told him that she would provide water for the camels before she gave him water to drink, 24:46, the exact version of the test, and that he gave her the presents after finding out about her lineage, 24:47.
When the servant ended his recounting of his meeting with Rivka, he demanded an immediate answer from the family whether they would agree to the match, 24:49. This need for an immediate response was because the effect of the supposed divination would start to diminish with time, and the family could begin to question the story.
The servant's plan worked perfectly. Rivka's brother and father, Lavan and Betuel, responded to the servant by stating that since the selection of Rivka was from G-d, they agreed to the match, 24:50,51. Note 24:50 first refers to Lavan and Betuel in the singular to indicate that each on his own believed that the match was from G-d.
This agreement meant that the servant had succeeded in his mission, and he left early the next morning, 24:54-61, before the family would change their mind and/ or learn that the test was staged. My guess is that the servant would have even wanted to leave immediately after he had the agreement, but it was too late at night to start the journey home.
Bibliography:
Leibowitz, Nehama (1905-1997), 1976, Studies in Bereshit, translated by Aryeh Newman, Jerusalem: The World Zionist Organization.
Perry, Menachem, 2007, Counter-stories in the Bible: Rebekah and her bridegroom, Abraham's servant, Prooftexts, 27, 2007, pp. 275-323.
Schein, Andrew, 1997, The test of Rebecca, Tradition: A journal of Orthodox Jewish thought, 31:4, pp. 28-33.