Would you say that Rivka came from a "nice family?"
But the title of this post is quite specific. The Rivka referred to is none other than Rivka Immeinu. Remember, Avraham sent his trusted servant, Eliezer, all the way back to his home country to find a suitable bride for Yitzchak. Eliezer spotted Rivka and came up with his own test to assess whether or not she was worthy of his master's son. He decided on her before asking who her parents were. While it's true that yichus is among the reasons offered for selecting a wife, (see http://kallahmagazine.blogspot.com/2008/08/15th-of-ave-take-4-differing-accounts.html) it seems to have been perceived only on the plus side rather than as a reason to screen someone out.
Yitzchak's marriage to Rivka is actually foreshadowed in the account of progeny that follows the story of the akeida. Avraham is told that his brother had children whose progeny lead up to the birth of his son’s bashert, Rvfka. A a number of people’s names are included in that genealogy, but the name of Lavan, Rifka’s older brother is conspicuously missing. One possible answer is that he need not be mentioned because once the Torah arrives at Bethuel’s name, it can immediately mention the birth of his daughter without listing his other children. However, the fact that her brother's name is left off shows that the standard of checking out the brother as a way of assessing if the family is "nice" should not be a deterrant to the match.
There is a principle advanced in Bava Basra (110A) that one who marries a wife should check out her brother because his character foretells how her sons will turn out. Rashi quotes this in connection to Vaera 6:23 to explain why the Torah mentions the name of the brother of Aharon’s wife. In that instance, the brother was a most illustrious character, a propitious choice for the husband. But we know that Lavan, Rivka's brother, was a notorious rasha, an idol worshipper who took pride in duping people. As for her father, according to the Midrash, the poison that he intended for Elizer was shifted to his own plate, so only his own death prevented him from carrying out a murder. It doesn't sound like a nice family, does it?
Nevertheless, Elizer presented Rivka as a suitable bride. And she proved herself a most worthy wife and matriarch. (As an added note, this parasha is layned on Rosh Hashana; perhaps it has something to teach about perspective and judgement.)
As for the argument that you don't just marry the individual, but the family the person was born into, that is very true. But knowing that his uncle was utterly wicked and not to be trusted, Yaakov, at the advice of his parents, sought an alliance with his daughter. Consequently, he had to contend with the trickery, suspicion, and threats of a father-in-law you would not wish on your worst enemy. Rachel and Leah also came from a family that would not make the "nice" cut in the shidduch question. Still, they were great in their own right and quite worthy of the role of Immahos.