Let G be an nonabelian has oder of 8. Let x be a character. Show that x(1)=1
Do you mean a multiplicative character (linear character) or a character of a representation? If you are referring the latter, it is not always true that x(1)=1.
For example, $\displaystyle \phi_8 \rightarrow GL_2(\mathbb{Re})$ can be a matrix representation (verify this), which maps the identity of $\displaystyle D_8$ to $\displaystyle 2 \times 2$ identity matrix in $\displaystyle GL_2(\mathbb{Re})$, i.e., $\displaystyle \chi(1)=2$, where $\displaystyle \chi$ is the character of $\displaystyle \phi$.