ANSWER: Before we answer what the Bible says about Hermaphrodites, we need to cover what the term means and where it came from. First, humans born with this condition have a genetic defect that gives them both female and male sexual characteristics and genital tissues. The reproductive organs of both sexes are present, in varying degrees, on the individual.
The term Hermaphrodite comes to us from pagan Greek mythology. Hermaphroditus was the son of the Greek gods Hermes and Aphrodite (the goddess of beauty, love and sex). According to the Roman poet Ovid, in his work entitled "Metamorphoses," a nymph named Salmacis falls completely in love with the handsome Hermaphroditus. Her passion is so strong that, after she physically clings to him, she asks the gods to make her one with him. The prayer is answered and the two bodies are fused together, forming one androgynous person.
One of the first known "Christian" references to this type of person comes to us from Augustine, who lived from 354 - 430 A.D. As one of the earliest Catholic theologians, he was recognized as a "saint" by the church. He wrote that, "As for the Androgyni, or Hermaphrodites, as they are called, though they are rare, yet from time to time there appears persons of sex so doubtful, that it remains uncertain from which sex they take their name . . ." (City of God, book 16, chapter 8).
Although Scripture does not directly address those who have this condition, it does give us some principles. The Apostle Paul's statements condemning homosexuality in Romans 1:26 - 27 shows that the ancient world could be just as confused about sex roles as in modern times. Such confusion is not what God desires. The Eternal's perfect will, delineated in the Bible, is that humans maintain a clear distinction between genders in their sex roles (Deuteronomy 22:5).
Parents who have a child born a hermaphrodite must face the difficult decision whether or not this condition should be altered surgically. It seems prudent that a variety of medical professionals are consulted and possible DNA (or other tests) be performed to determine which sex, genetically, dominates in the child with this condition.
Ideally, barring major hindering factors (e.g. high risk to the child's life, inaccessibility to qualified professionals that could perform specialized surgery, etc.), it might prove best for the child to be made the sex determine by their chromosomes (and not based on parental preferences of wanting to have a boy or a girl). Performing procedures to make hermaphrodites appear more like the sex that genetically dominates their bodies could help them grow up without a confused sense of gender identity.
According to a Wikipedia article on intersex humans, the World Health Organization (and other groups) have not taken a favorable view toward modifying the sexual characteristics of those, legally, too young to consent. Parents should be aware of any potential negative feedback they might receive due to their actions. They should also research or seek professional guidance regarding any laws that might hinder decisions they wish to make for a child born a hermaphrodite. As with any major decision (especially those affecting their children), parents should seek God's will.