A Lax-Friedrichs (LF) type of flux for a conservation law $\partial_tU+\partial_xf(U)=0$ is given by 
\begin{align}
F(U^-, U^+) = \frac{1}{2} \Big(f(U^-) + f(U^+)\Big)\cdot \nu - \frac{1}{2} \lambda(U^+ - U^-)
\end{align}
where for a classical LF flux in one dimension, $\lambda  = \Delta t / \Delta x$. I understand that the motivation comes from trying to write the central flux scheme 
\begin{align}
U^{n+1} = U^{n} - \frac{\Delta t}{\Delta x} 
\Big(f(U^n_{i+1}) - f(U^n_{i-1})\Big)
\end{align}
by 
\begin{align}
U^{n+1} = \frac{1}{2}\left(U^n_{i+1} + U^n_{i-1} \right)
- \frac{\Delta t}{\Delta x} \Big(f(U^n_{i+1}) - f(U^n_{i-1})\Big)
\end{align}
which then covers a wider domain than before and hence is more stable.
The step of replacing the value at a point by average seems like a small trick. I am curious what is the idea behind such scheme, and if we can arrive at the scheme by writing a modified version of PDE and proceeding along more natural path.