I am sorry I cannot comment your answer due to my low reputation. Gini and your suggested sparsity ratio ($l_1(x)/l_2(x)$) both give me the same value for $\lambda$. But

1. The problem I still see is that I cannot take into account how well the vector is solving the equation $Ax-y$. I would like to combine the residuum $l_1(A\hat{x}-y)$ and the sparsity $l_1(\hat{x})/l_2(\hat{x})$ into a new metric value. Do you know somethin from the literature?
2. Gini only works with positive values, but my measurement vector has lots of negative values. In my eyes this is a heavy disadvantage because many sparse vectors can have negativ values.

I am sorry I cannot comment your answer due to my low reputation. Gini and your suggested sparsity ratio ($l_1(x)/l_2(x)$) both give me the same value for $\lambda$. But

1. The problem I still see is that I cannot take into account how well the vector is solving the equation $Ax-y$. I would like to combine the residuum $l_1(A\hat{x}-y)$ and the sparsity $l_1(\hat{x})/l_2(\hat{x})$ into a new metric value. Do you know something from the literature?
2. Gini only works with positive values, but my measurement vector has lots of negative values. In my eyes, this is a heavy disadvantage because many sparse vectors can have negative values.