Maybe I am misinterpreting something, because according to my experiments, this function is **neither** convex nor concave.

The following is a counterexample (**EDIT:** I changed the example to use symmetric matrices):

\begin{equation*}
A=\begin{pmatrix}8 &4\\ 4 & 6\end{pmatrix},\quad B=\begin{pmatrix} 4 & 4\\ 4 & 6\end{pmatrix},\quad C=\frac{A+B}{2}=\begin{pmatrix}6 & 4\\ 4 &6 \end{pmatrix}.
\end{equation*}
For this choice, we have

\begin{equation*}
v(A) = (.7882, .6154),\quad v(B)=(.6154,.7882),\quad v(C)=(.7071, .7071).
\end{equation*}
But $\|v(C)\|_1 > 0.5\|v(A)\|_1 + 0.5\|v(B)\|_1$ (notice all there vectors have unit 2-norm as required).

A similar counterexample to potential concavity is also easy to find.

**EDIT 2:** Here is a counterexample to concavity.

\begin{equation*}
A = \begin{pmatrix}16&2\\ 2&16\end{pmatrix},\quad B = \begin{pmatrix}14&8\\8 &2 \end{pmatrix},\quad C = (A+B)/2
\end{equation*}
Then, we have
\begin{equation*}
v(A) = \begin{pmatrix}\tfrac{1}{\sqrt{2}}\\\tfrac{1}{\sqrt{2}}\end{pmatrix},\quad v(B) =\begin{pmatrix}\tfrac{2}{\sqrt{5}}\\\tfrac{1}{\sqrt{5}}\end{pmatrix},\quad v(C)= \begin{pmatrix}\tfrac{3+\sqrt{34}}{\sqrt{68+6 \sqrt{34}}}\\\tfrac{5}{\sqrt{68+6 \sqrt{34}}} \end{pmatrix}.
\end{equation*}
Doing the numerics with this shows that $\|v(C)\|_1-0.5(\|v(A)\|_1+\|v(B)\|_1) = -0.0150285...$.