Regularized Least Squares by Laplacian Operator - Optimal Value of the Regularization Factor (Lagrangian Multiplier)
Let me rewrite the problem:

$$ \arg \min_{x} \frac{1}{2} {\left\| A x - y \right\|}_{2}^{2} + \frac{\lambda}{2} {\left\| L x \right\|}_{2}^{2} $$

In order ot say something intelligent about the optimal value of $ \lambda $ one must set an optimization criteria.
Since you don't have one there is no well defined answer.

What might assist you is reformulating the problem in an equivalent form:

$$\begin{align*}
\arg \min_{x} \quad &amp; \frac{1}{2} {\left\| A x - y \right\|}_{2}^{2} \\
\text{subject to} \quad &amp; {\left\| L x \right\|}_{2}^{2} \leq \alpha &amp; 
\end{align*}$$

Both forms are equivalent in the sense that for any $ \alpha $ of the 2nd form there exists $ \lambda $ such that both solutions are the same.

Now if you can say something about the smoothness level of the image which will let you set $ \alpha $ intelligently than you'll be able to extract $ \lambda $ to match it.

Pay attention that the connection isn't trivial and depends on the data itself among other things (See my answer to Significance of λ in Basis Pursuit).