Dede Mirabal's husband, Jaimito, brands his three sons with his own name. Their first names are all Jaime as well, so they have to go by their middle names--Enrique, Rafael, and David. This is a symbol of Jaimito's way of thinking. He sees his sons (and also his wife) as "his" property. He sees himself, the father and the husband, as the head of the household, and thus as the sole decision-maker.
Jaimito's selfishness and various flaws are a huge part of the reason that Dede is the only Mirabal sister who does not join the revolution against Rafael Trujillo, the cruel dictator of the Dominican Republic. The husbands of Minerva, Patria, and Maria Teresa all become involved in the revolution in some way or other. But Dede's husband, through practicality and cowardice, refuses to become part of the underground movement. Thus, he also does not allow Dede to do so, because he thinks that the decision of the husband/father is final.
At one point in the book, Dede comes home to find that her three boys are gone. Panicked, she thinks that Jaimito has taken them away from her for trying to help her sisters. In fact, he did take them to their grandmother's house in order to purposefully make Dede feel terrified. Jaimito's sense of ownership over their lives is very evident.
Although Dede eventually divorces Jaimito, his iron fisted rule over her household has a deep effect on her life. After the murders of her three sisters (who are on their way to visit their three husbands, who are all in jail for being revolutionaries), Dede must live with the profound and complex grief and survivor's guilt for the rest of her life. She ultimately blames herself for being too much of a coward to defy Jaimito and join the revolution.
Thus, it's pretty clear that Jaimito's proprietary names for his children are symbolic of his close-mindedness.