2 slight modification and explanation added

I do not think much/anything can be done.

Let us leave the simple special cases of rank $M$ equal $0$ or $1$ aside.

So, an example of a $n$ times $n$ rank two matrix $M$ such that the rank of $|M|$ is full:

Take the two vectors $e=(1, \dots, 1)$ and $u = (0, -1, -2, \dots, -(n-1))$. Consider the matrix $M$ formed by $e$ and $(j+ 0.5)e je + u$ for $j=0, \dots n-2$.

The absolute value $|M|$ has full rank since the line for any $j\ge 1$ is $je+u + v_j$ where $v_j = (2 \max(0, (i-1)-j) )_i$. and thus has exactly the first $j+1$ coordinates equal to $0$. So, we get $e$, and $u= - |u|$ and all the $v_j$ for $j=1, \dots, n-2$ in the spanned space, and these $n$ are independent.

Variations of this should give (all?) kinds of intermideate phenomena.

(This Edit: slight change and explanation; perhaps the orginal would also work but the present example seems clearer and was written a bit quickly hope the 'real' original, which I did not misunderstand something or make an error.thought I should modify while typing for some dubious reasons. Sorry for the edit-noise.)

1

I do not think much/anything can be done.

Let us leave the simple special cases of rank $M$ equal $0$ or $1$ aside.

So, an example of a $n$ times $n$ rank two matrix $M$ such that the rank of $|M|$ is full:

Take the two vectors $e=(1, \dots, 1)$ and $u = (0, -1, -2, \dots, -(n-1))$. Consider the matrix $M$ formed by $e$ and $(j+ 0.5)e + u$ for $j=0, \dots n-2$.

Variations of this should give (all?) kinds of intermideate phenomena.

(This was written a bit quickly hope I did not misunderstand something or make an error.)