After this acceptance, the servant gave gifts to Rivka, her mother and her brother, but not to her father, 24:53. The following morning, Rivka's mother and brother tried to delay the marriage by requesting that Rivka stay at home for some time, and again Betuel is absent, 24:55. Where was Betuel in the morning and why did the servant not give him presents?
Rashi (on 24:55) quotes from the Midrash that Betuel died during the night, while the Bekhor Shor (on 24:55, also see Luzzatto on 24:55) explains that he did not speak in the morning since he did not want to delay the marriage.
Perry (2007) suggests that it is possible to explain Betuel's disappearance based on idea that Rivka's mother and father had separate houses. He suggests that Rivka was always in her mother's house (24:28), and that Lavan went back and forth between her mother's house and her father's house. When Rivka came home after meeting the servant, she went to her mother's house, but the servant went with Lavan to her father's house. After Lavan and Betuel agreed to the match, the servant went back to the mother's house, where he gave presents to Rivka and to Lavan who went with him but Betuel stayed in his house so he did not get any presents. Finally, in the morning, when the servant went to take Rivka, he went to Rivka's mother's house and Betuel was not there.
While these approaches are possible, I do not understand why there is even a question why Betuel did not speak in 24:55. After Betuel heard the servant's tale he said that the marriage was destined by G-d, and that "he would not say anything bad or good," 24:50. This means he would keep quiet. Why should one expect him to say something the following morning?
One might then wonder why 24:55 records that Rivka's brother asked for Rivka to stay at home for some time if Lavan had also said in 24:50, that "he would not say anything bad or good"? Did Lavan go back on his word? More likely, the reference to Rivka's brother in 24:55 is not to Lavan but to a different brother of Rivka. The proof is that had 24:55 been referring to Lavan, then the text would have referred to him by his name, as occurs throughout the chapter. Instead, 24:55 is referring to another brother of Rivka who is referred to anonymously as Rivka's brother, and this anonymous brother asked for Rivka to remain at home for some time.
With this understanding, maybe we can also understand why the servant did not give any gifts to Betuel in 24:53. The servant already had the agreement from Lavan and Betuel for the match, and they had indicated that they did not want to have anything more to do with the servant as they said, "here is Rivka before you, take her and go," 24:51. Thus, the servant did not give them any gifts. However, the servant must have sensed that not everybody else in the family was so thrilled with the match. Thus, he gave Rivka's mother and her anonymous brother (not Lavan) gifts to get them to be more supportive of the match, but in this effort he did not succeed since the following morning they tried to stop the match by requesting that Rivka not leave right away, 24:55.
Bibliography:
Fox, Everett, 1995, The Five Books of Moses: A new translation, New York: Schocken Books.
Perry, Menachem, 2007, Counter-stories in the Bible: Rebekah and her bridegroom, Abraham's servant, Prooftexts, 27, 2007, pp. 275-323.