It is not clear to me what the level of difficulty of this question is, but it appears to involve notions number theoretic randomness, and so it seems like a reasonable candidate for an MO question. 

The objective is to obtain an asymptotic formula for the sum 

$$S(X)=\sum_{q\leq X}\frac{1}{\phi(q)}\sum_{\chi}L^2(0,\chi),$$

and it can easily be show that determining this is equivalent to knowing something interesting about Farey fractions. It is fairly elementary that 

\begin{eqnarray}\label{}
S(X)=\sum_{q\leq X}\frac{1}{\phi(q)}\sum_{\chi} \left(  \sum_{a\leq q} \chi(a)\zeta\left(0 ,\frac{a}{q}\right)   \right)^2
\end{eqnarray}

where $\zeta(s,a)$ is the Hurwitz zeta function so, denoting by $a^*$ the multiplicative inverse of $a$ modulo $q$, expanding out the square and using the orthogonality relations for the Dirichlet characters, one obtains

$$S(X)=\sum_{q\leq X}\sum_{(a,q)=1}\left(\frac{a}{q}-\frac{1}{2}\right)\left(\frac{a^*}{q}-\frac{1}{2}\right).$$ 

Noting that the mean value of the non-trivial Farey fractions ($0$ and $1$ being trivial) is $1/2$ for all $X$, at this point one probably realises that $S(X)/X^2$ is proportional (by a factor of $3/\pi^2$) to the correlation of the non-trivial Farey fractions of order $X$ with their multiplicative inverses modulo their denominators. 

Since one would expect the values of $a/q$ and $a^*/q$ to be independent as $q\rightarrow\infty$, one would expect that $$S(X)=O(X^{1+\epsilon})$$ and  so I would like to pose the following question:

>Can an asymptotic formula for $S(X)$ be determined, or just a non-trivial upper or lower bound, via the Farey fractions or the Dirichlet L-functions?  

A potential way to proceed is to multiply out the product and use the symmetry about $1/2$ to obtain 

$$S(X)=-\frac{1}{2}+\sum_{q\leq X}\sum_{(a,q)=1}\left(\frac{aa^{*}}{q^2}-\frac{1}{4}\right).$$
Using the fact that $aa^{*}-k(a)q=1$, where $k(a)$ is the greatest number of integer multiples of $q$ less than $aa^{*}$, and $k(a)/q$ is another non-trivial Farey fraction, one obtains 

$$S(X)=-\frac{1}{2}+\sum_{q\leq X}\sum_{(a,q)=1}\left(\frac{k(a)}{q}-\frac{1}{4}\right) +O(\log X).$$ 
However, the mapping $a\rightarrow k(a)$ is not an automorphism of the  multiplicative group of integers modulo $q$ and the behaviour of this mapping seems to be rather complex.