My wife and I have argued this, and I've had trouble finding the answer. Which sex carries the genes to determine the sex of the offspring... basically, which parent decides the gender of the baby?
The sex of a human baby is determined by the composition of its sex chromosomes (a single distinct pair among humans' 23 pairs of chromosomes). Females possess two copies of the same chromosome (referred to as the 'X' chromosome); males have one copy of the X chromosome and one copy of the smaller, hook-shaped Y chromosome.
When fertilization occurs, the new gamete (the initial cell from which a fetus grows) always inherits one of the mother's X chromosomes, and either a X or a Y from the father, depending on which chromosome the fertilizing sperm cell happened to inherit. One could say, then, that the father—or, at least, his sperm—determines the sex of the child. On the other hand, the first sperm to reach the egg isn't necessarily the one that fertilizes it; human eggs are rather choosy about that sort of thing. So, in an indirect way, the maternal parent also has some influence on the sex of the child.
There are some genetic and developmental disorders which disrupt the normal activity of the sex chromosomes. Men with Klinefelter's Syndrome, for example, have an extra copy (or copies) of the X chromosome; females with Turner's Syndrome have only one X chromosome. Abnormal levels of sex hormones (such as testosterone) during embryonic development may also cause genetically XX fetuses to develop as males, or genetically XY fetuses to develop as females. Individuals with these kinds of conditions are almost always sterile, and often suffer mild to severe developmental abnormalities.
For more information on the workings of sex chromosomes and sexual development, check out our encyclopedia article on sex. I hope this helps you resolve your dispute.