Mendel's law of segregation simply states that different traits are determined independently of each other. That means that the genes for the child's eyes are inherited irrespective of those determining the nose, and the hair color. This means that just because the child's eyes match his dad's and his nose matches mom's doesn't influence the hair color.
Mendel also found that some genes are expressed over others. The gene that gets expressed over another in a pairing is called the Dominant gene and the other gene is called a Recessive Gene. For example the gene for tall pea plants is dominant to the gene for short pea plants. So if you cross tall plants with short plants Generation 2 will all be tall, as they all have the dominant tall gene. It will only be in Generations 3 and later, that the recessive gene for short plants will be expressed.
The same goes for hair color. The paternal Grandmother (Dad's mother) had two recessive genes for her hair color. The paternal Grandfather must have had a dominant gene that his son (the Dad) inherited determining his hair color, but because Grandmother had only recessive genes, Dad inevitably received the recessive gene for the other hair color. Mom must also have carried the same recessive gene, and this child just happened to receive both recessive genes, thus getting his grandmother's hair color.