Rashi says that she went to the Beis Medrash of Sheim, and in the next passuk Rashi brings that Hashem answered her through the nevuah of Sheim. This is in the Medrash Rabbah here 63:6, and 20:6, 45:10, and 48:20, and sort of in the Yerushalmi in Sotah right in the beginning of seventh perek, and the Zohar here on this passuk. Some say to Sheim, some say through a malach.
According to those that say she went to Sheim or Eiver, why did Rivka go to Sheim and Eiver with her problem? Why didn’t she just ask her own husband, Yitzchak? What about Avraham, who lived until Yakov and Eisav were 15 years old? Was there something about the nature of her question that required that she not go to them, and instead go to Sheim?
(The Ibn Ezra, as usual, solves the problem by disregarding what Chazal say: he says that she did go to Avraham. The Abravanel also says this as an alternative to the various Medrashim. This is, of course, a valid interpretation of the passuk, but it does nothing to explain what Chazal were thinking.)
The Gur Aryeh (and, I'm told, the Baal Haturim in his pirush) says that she was afraid that the problem she perceived in her child arose from, and reflected, her own flaws, and she was ashamed to go to her husband. .
An acquaintance of mine suggested that the fact that Yitzchok learned in the Beis Medrash of Sheim and Eiver showed that they were the gedolei hador, so they were first people to consult. Another friend vehemently disagreed, saying that there is no question that Yitzchok, and of course Avraham, were far greater than Sheim and Eiver, and the fact that Yitzchak left home to learn there proves nothing.
The Ramban on Breishis 27:4 discusses this tangentially. He asks, when Rivka overheard Yitzchak tell Eisav that he would give him the brachos, she came up with a ruse to subvert Eisav's plan. Why didn’t she simply tell Yitzchak about what she heard from Sheim, that “ve’rav ya'avod tza’ir,” which meant that Eisav would serve Yaakov, and which would mean that the brachos should go to Yaakov? The Ramban answers "Apparently Rivkah never told him [Yitzchok] about G-d's prophecy to her, "and the older shall serve the younger" (Breishis 25:23), for if she had, how could Isaac go against the word of Hashem? At first, she did not tell him because of her sense of morality and modesty, for "she went to inquire of Ha’elokim" (Breishis 25:22), and she had gone without Isaac's permission; or perhaps she thought, "I need not report a prophecy to a prophet, for he is greater than he who told me," [Sheim, according to the Medrash.] And now she did not want to say to him, "I was told such and such by the Lord before I gave birth," for she reasoned, out of [Yitzchak's] love for [Eisav], [Yitzchak] would not bless Yakov, but would leave everything in Hashem's hands; and she knew that for this reason [giving Jacob cooked food that tasted like game] Yakov would receive the blessing from his mouth with a full heart and a willing soul.” (Then the Ramban says something vague about hidden intentions of Hashem.) So: from the Ramban we see several possiblities: 1. that either she didn't want to tell Yitzchak what Shem told her because she went to consult Shem without Yitzchak's permission; or 2. that she figured that if the lesser navi (Shem) knew something, then certainly the greater navi (Yitzchak) knew it as well, (and 3. if he didn't, then there must be a reason Hashem did not want Yitzchak to know about it).
The Chizkuni says this as well-- that Rivka never told Yitzchak what she was told, and that she must have been told to not tell him, or at least was not told to tell him. This, too, is a post facto validation of her decision to seek the answer outside her home.
I would say, expanding on the Ramban and the Chizkuni, that the three points we elicited from the Ramban are not either/or. They all are true. Rivka reasoned that if her husband or father in law knew what was going on, they would have told her. From their silence she deduced two things: that they didn't know, and that Hashem didn't want them to know. So she make an independent decision and went elsewhere, thinking that even if Hashem didn't want Yitzchak and Avraham to know, Hashem might tell her. As it turned out, she was 100% correct.
The Netziv here also deals with the question. He says there are two types of nevuah, two types of nevi'im. One is a navi to whom Hashem speaks; the other is one who sees hidden things through ruach hakodesh, like Shmuel, the "Ro'eh," to whom people would go to find out where their missing donkeys had gone. As is evident from Shmuel, these two types of Nevu'ah are not mutually exclusive; but the Netziv says that Avraham was a type one navi only, while Shem was type 2, and that’s why Rivka went to him. She needed a see-er, a Ro'eh, an “Adam Gadol she’yad'ah ki hu ro’eh veyodei’ah.”
Rabbi Kenny Nieman said that she went to Sheim because he was the zakein— Sheim lived 600 years, and he was probably around 550 at that time. Rivka's husband and father in law were, comparatively, youngsters. The distinctive wisdom of a zakein, especially a zakein who was a navi for 400 years, is the greatest possible resource. This teretz reminded me of the dinner for MTJ where they were mechabeid Reb Moshe, and when Reb Yakov Kaminetski spoke, he said that it might seem to be gaivah for him to evaluate Reb Moshe, but there cannot be any doubt that he (Reb Yaakov) is the zakein, and so he had a right to state his opinion of Reb Moshe’s gadlus.
Thank you, Eli, for finding the Medrash Seichel Tov, (authored/compiled by R. Menahem Ben Shlomo, Italy (?), in 1139) that says exactly this:
זקנים ללמדך שכל המקבל פני זקן שבדור, כאילו מקבל פני שכינה
Reb Berel Povarsky has a very nice discussion about this kashe. He asks this, and many other questions, such as, didn’t she know that her husband had a bad brother, Yishmo’eil, and that there was a process of ‘zikuch’ before the 12 shvatim could be born? Also, Chazal said that Shem showed her Rebbi and Antoninus. How does a good descendant console her for a son who is a rasha? Answer— she knew everything, but wondered, why did the zikuch by Avrohom take place through his pilegesh, Hagar, and now it is taking place through me— lamah zeh anochi— why am I the one that has to bear the bad one. This, of course, was not a question she could ask her husband’s family, because of the concern that the cause was some personal flaw she carried, similar to the Gur Aryeh. The answer that Shem gave her was that Yishma’eil, being the zikuch son, and having been born by the pilegesh, had no redeeming qualities, and is and will always be a pereh adam. (But see above end of Chayei Soroh that Yishmo’eil did tshuvoh before he died, and Eisav never did.) But Eisav needed to have a higher character, and not just be an outlaw, and so you were the one to bear him and to infuse into him these higher qualities. This is why she was shown Rebbi and Antoninus; to show her both the benefit Klal Yisrael will have from him, and also the more refined character a ben Eisav is capable of.
And here is what I think: that she didn’t go to Sheim, but rather to his Beis Medrash. She wanted to ask Hashem herself, or to be mispallel, and to do so required that she go to a makom kadosh, just as Chana went to the Bais Hamikdash when she wanted to ask Hashem for children. And just as Chana was then answered through Eli, the Kohen Gadol who was there at the time, Rivka’s question/tefilla was answered through Sheim, who was a kohen gadol (Nedarim 32b and Targum Yerushalmi Breishis 14:18), or his descendant, Ever, who was in the Beis Medrash at that time. She could have asked her husband or her father in law, but the more appropriate response to a challenge is to attempt to resolve the question yourself, and not to abdicate the opportunity for personal growth by passing off the issue to someone else. Even if Malkitzedek lost the Kehuna for giving a bracha to Avraham before thanking Hashem, it doesn't really matter. He was the Gadol in a place that was meyuchad for Limud Hatorah and Hashra'as Hashechina.
The word "lidrosh" here means what it means in ישעיהו נה where it says דרשו ה' בהמצאו . The Gemara in Rosh Hashanna 18a says that בהמצאו means either in the setting of a tzibbur, because the Shechina rests on a tzibbur, or during Aseres Yemei Teshuva, when the Shechina makes itself available. The same בהמצאו applies to a place of kedusha. It may be negi'us, but I think my pshat is peshuto shel mikra.
Having said this, it becomes clear that this is what the Medrash Seichel Tov that I brought above means:
What's the lesson here, then? The lesson is this: That a person, certainly one who has problems having or raising children, but also anyone that faces a life difficulty, should do as Channah and Rivka did. Go to the presence of a zakein who is a tzadik and a great talmid chacham, and be mispallel to Hashem for help with your problem. Certainly, seeing a bracha is wise. As the Gemara in Bava Basra 116a says,
If someone else has already said this, I don't want to know about it, thank you. I know that the Ramban says that 'lidrosh' means tefilla, and that the Abravanel argues. But what I am saying, that Rivka went to a place where there was Hashra'as Hashechina, and connecting it to the story of Chana and how she was answered, and the Gemara in Rosh Hashanna, and the Passuk in Yeshaya, is far more comprehensive than what the Ramban says, especially since unlike the Ramban who is arguing with Rashi, what I'm saying explains Rashi. So don't tell me that it's mefurash in the Ramban. It's not mefurash in the Ramban and it's befeirush not like the Ramban.