Note that if $a$ is not an eigenvalue of the matrix $A$, then the matrix $A-aI$ is nonsingular, that is, the null space of $A-aI$ is the zero space, and the dimension is $0$.
Thus to obtain a nonzero point, your favorite number should be an eigenvalue.

Let us find the eigenvalue. The characteristic polynomial $p(t)$ of $A$ is
\begin{align*}
p(t)&=\det(A-tI)\\
&\begin{vmatrix}
5-t & 2 & -1 \\
2 &2-t &2 \\
-1 & 2 & 5-t
\end{vmatrix}\\
&=(5-t)\begin{vmatrix}
2-t & 2\\
2& 5-t
\end{vmatrix}-2\begin{vmatrix}
2 & 2\\
-1& 5-t
\end{vmatrix}+(-1)\begin{vmatrix}
2 & 2-t\\
-1& 2
\end{vmatrix}\\
&\text{(By the first row cofactor expansion.)}\\
&=(5-t)(t^2-7t+6)-2(-2t+12)-(-t+6)\\
&=-t^3+12t^2-36t\\
&=-t(t-6)^2.
\end{align*}

Eigenvalues are roots of the characteristic polynomial.
Thus eigenvalues of $A$ are $0$ and $6$, with algebraic multiplicities $1$ and $2$, respectively.
Since the matrix $A$ is a real symmetric matrix, it is diagonalizable. Thus the geometric multiplicity of each eigenvalue is equal to its algebraic multiplicity.
Therefore the dimension of the null space of $A-6I$, which is the geometric multiplicity of the eigenvalue $6$ is equal to $2$, and the dimension of the null space of $A$ is $1$.

Thus, if you take $a=6$, you get $2\times 5=10$ points. If your favorite number is $a=0$, then your score is $1\times 5=5$ points. The other choice of $a$ will be zero points.

Comment.

If you didn’t notice that the given matrix is a real symmetric matrix, then you need to find eigenvectors corresponding to eigenvalues $6$ and to show that the eigenspace has dimension $2$.

How to Diagonalize a Matrix. Step by Step Explanation.
In this post, we explain how to diagonalize a matrix if it is diagonalizable.
As an example, we solve the following problem.
Diagonalize the matrix
\[A=\begin{bmatrix}
4 & -3 & -3 \\
3 &-2 &-3 \\
-1 & 1 & 2
\end{bmatrix}\]
by finding a nonsingular […]

Quiz 13 (Part 1) Diagonalize a Matrix
Let
\[A=\begin{bmatrix}
2 & -1 & -1 \\
-1 &2 &-1 \\
-1 & -1 & 2
\end{bmatrix}.\]
Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$.
That is, find a nonsingular matrix $A$ and a diagonal matrix $D$ such that […]

Quiz 13 (Part 2) Find Eigenvalues and Eigenvectors of a Special Matrix
Find all eigenvalues of the matrix
\[A=\begin{bmatrix}
0 & i & i & i \\
i &0 & i & i \\
i & i & 0 & i \\
i & i & i & 0
\end{bmatrix},\]
where $i=\sqrt{-1}$. For each eigenvalue of $A$, determine its algebraic multiplicity and geometric […]

Given All Eigenvalues and Eigenspaces, Compute a Matrix Product
Let $C$ be a $4 \times 4$ matrix with all eigenvalues $\lambda=2, -1$ and eigensapces
\[E_2=\Span\left \{\quad \begin{bmatrix}
1 \\
1 \\
1 \\
1
\end{bmatrix} \quad\right \} \text{ and } E_{-1}=\Span\left \{ \quad\begin{bmatrix}
1 \\
2 \\
1 \\
1
[…]