UPDATE: Because I was hoping that state the question as concisely as possible, the original post did not include a precise definition of arithmetic 3-manifold only a reference to Maclachlan and Reid's book where it is defined. However, this lead to an ambiguity that caused conflicting answers. With the included definition, Ian Agol's answer is correct. However, given a more restrictive definition of arithmetic 3-manifold, which is (less standard, but) used in a paper of Gromov and Guth, John Pardon's answer is correct. I apologize for the confusion and appreciate the thoughtful responses of the MO community.

Following a definition was taken from Maclachlan and Reid, ``The Arithmetic of Hyperbolic 3-manifolds'', Chapter 8.2:

Let $k$ be a number field with exactly one complex place and $A$ a quaternion algebra over $k$ which is ramified at all real places. Furthermore assume $A\subset M_2(\mathbb{C})$ via a complex embedding of $k$. If $\mathcal{O}$ is an order in $A$ and $\mathcal{O}^1$ is the set of elements of $\mathcal{O}$ of determinant 1, then a subgroup $\Gamma$ of $SL(2,\mathbb{C})$ is *arithmetic* if it is commensurable with $\mathcal{O}^1$ and a subgroup $\Gamma \subset PSL(2,\mathbb{C})$ is *arithmetic* if it is commensurable with $P\mathcal{O}^1$.
If we consider the natural action of $PSL(2,\mathbb{C})$ on the upper-half space model of $\mathbb{H}^3$, a finite volume hyperbolic 3-manifold (or 3-orbifold) $Q \cong \mathbb{H}^3/\Gamma$ is *arithmetic* precisely when $\Gamma$ is arithmetic.

Arithmetic 3-manifolds tend to be atypical many ways. For example, one non-generic property of arithmeticity observed by A.Borel in "Commensurability classes and volumes of hyperbolic 3-manifolds" is the following: for a fixed $V>0$, there are only finitely arithmetic 3-manifolds of volume less than $V$.

Along those lines, I was wondering about the following:

Question: For a fixed Heegaard genus $g$, are there only finitely many arithmetic manifolds of Heegaard genus $<g?$