Question

How does a person get a trait that neither parent exhibits?

Answer 1

each gene has two alleles - paternal and maternal, of which they transfer only one. homozygotes - two copies of the same allele. heterozygotes- different allelles. when two non identical allelles are together - the one expressed is called a DOMINANT, the non expressed - recessive.
lets think about a family with parents with brown eyes
BROWN blue XXXXX BROWN blue

their offspring will have either one of these four comnibations:
BROWN BROWN
BROWN blue
blue BROWN
all of which will be brown btw

and blue blue - which will have blue eyes

voila

Answer 2

phenotypically it is possible but genotyipically its not.
we can not get new gene which our parents do not have