We introduce and study the locus $\mathbb{M}_{g,d}^\textrm{nd}$ of genus $g$ tropical plane curves of gonality $d$ inside the moduli space $\mathbb{M}^{\textrm{nd}}_{g}$ of tropical plane curves of genus $g$. Each such tropical curve arises from a Newton polygon, and we conjecture that the gonality of the tropical curve is equal to an easily computed parameter of this polygon called the expected gonality, closely related to the lattice width of the polygon. Let $\mathbb{M}_{g,{\underline{d}}}^\textrm{nd}$ denote the locus of tropical curves whose associated Newton polygon has expected gonality $d$. We prove that for fixed $d$ and sufficiently large genus $g$, the dimensions of these two loci agree: \[ \\dim\left(\mathbb{M}_{g,d}^\textrm{nd}\right) =\dim\left(\mathbb{M}_{g,{\underline{d}}}^\textrm{nd}\right). \] Our results provide evidence that, in sufficiently high genus compared to expected gonality, the gonality of a tropical curve is determined by the expected gonality of the Newton polygon from which it arises.