Itenberg, Kazarkov, Mikhalkin and Zharkov have formulated a definition of tropical $H^{p,q}$ and presented it at a number of conferences, although I don't think that there is a preprint yet. You can watch Zharkov's talk on this from 2009 here.

Their definition is restricted to the case that $\mathrm{Trop} \ X$ locally looks like a tropical linear space. (For example, if $X$ is a curve then, at a vertex of degree $d$, the directions of the incoming edges must span a space of dimension $d-1$ and the unique relation between the minimal lattice vectors on these directions must be that their sum is $0$.) This can be thought of as a tropical "smoothness" condition.

Roughly speaking, $H^{0,q}$ is related to the topological cohomology of $\mathrm{Trop} \ X$ which should, in this context, be viewed as the cohomology of the sheaf of locally constant functions on $\mathrm{Trop} \ X$. $H^{p,q}(X)$ is related to the cohomology of a sheaf on $\mathrm{Trop} \ X$ which is related to the degree $p$ part of Orlik-Solomon algebras of the matroids locally describing the relevant tropical linear spaces. (In other words, to $H^p$ of the corresponding hyperplane arrangement complement.)

Until a preprint arrives, the best reference seems to be the papers of Kristin Shaw, a student of Mikhalkin. See Section 2.2 of her thesis for a good start.