I'll assume that your hyperbolic orbifold $\mathcal{O}$ has one boundary component (if you'd like to rephrase the question to apply to multiple boundary components, then I can revise my answer), and that its interior admits a complete hyperbolic metric (which is the usual meaning of a hyperbolic orbifold). Using the orbifold theorem, one may assume that the interior of $\mathcal{O}$ is geometrically finite and has no cusps (if $\mathcal{O}$ is finite volume with a single cusp, then I think it is trivially true, since the rank of Euclidean orbifolds are no more than 3).

Under these assumptions, the answer to your question is yes. The ranks of hyperbolic 2-orbifolds were computed by Zieschang et al. If $\partial\mathcal{O}$ has genus $g$ and $p$ cone points, then they show that $rank(\pi_1\mathcal{O})\geq 2g+p-1$ for generic orbifolds, and $rank(\partial\pi_1(\mathcal{O}))=p-2$ for a particular family with $g=0$ and $p$ even and $\geq 4$. So we need to show $rank(\pi_1(\partial\mathcal{O}))\leq g+p/2-1$ generically. In the case $p=0$, one has $rank(\partial\pi_1(\mathcal{O}))=2g$.

A theorem of Sullivan shows that the deformation space of geometrically finite structures on $\mathcal{O}$ is parameterized by the Teichmuller space of $\partial{O}$. This follows from the theory of quasiconformal deformations of Kleinian groups. Now, one follows the proof of the Ahlfors finiteness theorem. If $rank(\pi_1\mathcal{O})=k$, then the space of deformations of representations of $\pi_1\mathcal{O}$ into $PSL_2(\mathbb{C})$ has $\mathbb{C}$-dimension $\leq 3k-3$ (this follows by computing the dimension of the variety of representations, and using that the conjugacy action is faithful since the generators are non-commuting). On the other hand, the Teichmuller space of $\partial\mathcal{O}$ has complex dimension $3g-3+p$, so we get $3g-3+p\geq 3k-3$, or $g+p/3\geq k$. From the rank computation above, then we get for $p\geq 3$, $\frac12 rank(\pi_1\mathcal{O})\geq g+p/2-1/2 \geq g+p/3 \geq k$. In the other cases, and when $g=0$, one can analyze the inequality case-by-case.

One may obtain an estimate improving your factor of 4 to a factor of 3.

The ranks of hyperbolic 2-orbifolds were computed by Zieschang et al. If $\partial\mathcal{O}$ has genus $g$ and $p$ cone points, then they show that $rank(\pi_1\partial\mathcal{O})\leq 2g+p-1$, except in the case $p=0$, one has $rank(\partial\pi_1(\mathcal{O}))=2g$ (of course, one may deduce this estimate directly by thinking about the punctured case). The same argument (half-lives, half-dies) applies in that case (as Igor observes), so I'll assume $p>0$.

A theorem of Sullivan shows that the deformation space of geometrically finite structures on $\mathcal{O}$ is parameterized by the Teichmuller space of $\partial{\mathcal{O}}$. This follows from the theory of quasiconformal deformations of Kleinian groups. Now, one follows the proof of the Ahlfors finiteness theorem. If $rank(\pi_1\mathcal{O})=k$, then the space of deformations of representations of $\pi_1\mathcal{O}$ into $PSL_2(\mathbb{C})$ up to conjugacy has $\mathbb{C}$-dimension $\leq 3k-3$ (this follows by computing the dimension of the variety of representations, and using that the conjugacy action is faithful since the generators are non-commuting). This is also the dimension of the space of geometrically finite reps., since these are structurally stable.

On the other hand, the Teichmuller space of $\partial\mathcal{O}$ has complex dimension $3g-3+p$, so we get $3g-3+p\leq 3k-3$, or $g+p/3\leq k$. From the rank computation above, then we get $\frac13 rank(\pi_1\partial\mathcal{O})\leq 2g/3+p/3-1/3 \leq g+p/3 \leq k$. Obviously the worst estimate holds when $g=0$. One might be able to improve this result taking into account the relators.