In Gromov-Witten theory, why is the string coupling constant weighted by $2g-2$?

Here is a purely mathematical reason why we prefer to put $u^{2g-2}$ in our generating function instead of, say, $u^g$.

The generating function you write, $$\sum_{g,\beta} GW_{g,\beta}(X) u^{2g-2}Q^{\beta},$$ is the generating function for the connected Gromov-Witten invariants of $X$ (I've thrown in another variable that tracks the class $\beta$). Connected here means that the invariants are obtained from the moduli space of maps with connected domains. This generating function is usually called $F$ because we reserve the letter $Z$ for the disconnected generating function:

$$Z = \exp \left(\sum_{g,\beta} GW_{g,\beta}(X) u^{2g-2} Q^{\beta}\right)$$

The coefficients of this generating function are the disconnected Gromov-Witten invariants -- i.e. the invariants obtained from the moduli space of maps with (possibly) disconnected domains.

In order for the relationship between the connected and disconnected invariants to be $Z=\exp(F)$, we need that the quantity tracked by the variable to be additive under disjoint union and so we prefer the Euler characteristic $2g-2$ to the genus $g$ ($g$ is not additive under disjoint union). For disconnected invariants, Euler characteristic is much more natural than genus (which would have to be defined via Euler characteristic anyway). Note that much of the interesting features in GW theory are formulated via the disconnected invariants (e.g. the MNOP conjecture).

Here are two ways how physicists think about the string coupling constant:

1) In usual quantum field theory defined by quantization of a classical field theory, the partition function is defined by a path integral of the form $Z= \int D\phi e^{\frac{S(\phi)}{\hbar}}$ where $S$ is the classical action. In particular, in the classical limit $\hbar \rightarrow 0$, we expect a localization around the classical solution $\phi_{cl}$ to classical equation of motion, and so a behaviour like $Z \sim e^{\frac{S(\phi_{cl})}{\hbar}}$, i.e. $F=log(Z) \sim \frac{S(\phi_{cl})}{\hbar}$. Perturbative expansion in $\hbar$ will give a pertubative expansion of $F$. Standard Feynman diagram tecnhiques show that this expansion is of the form $F \sim \sum_{L} F_L \hbar^{L-1}$ where $F_L$ receives contribution from (connected) Feynman diagrams with $L$ loops.

We expect an open string theory to reduce at low energy to an usual quantum field theory with an expansion as above. But the closed string coupling $g_s$ is related to the open string coupling $\hbar$ by $g_s \sim \hbar^2$: this follows by unitarity from the annulus diagram which can be interpreted either as a one loop open string or a tree level closed string.

2) The point 1) is a spacetime point of view. There is also a worldsheet point of view. The non-linear sigma model is a two dimensional theory of maps from a two dimensional $\Sigma$ to a space $X$. The action of such theory has to be local and one natural term with this property is given by the integral of the curvature of $\Sigma$, which by Gauss-Bonnet is given by the Euler characteristic $2-2g$. This origin of the appearance of the Euler characteristic is essentially the one mentioned in the answer given by Jim Bryan: additivity gives a way to reconstruct a global quantity from a local one.

The physical explanation is in large N gauge theory. The Feynman diagram you are computing has become a ribbon graph. You see how the vertices, edges and faces are scaling as you increase the rank of the lie algebra. This is the reasoning for why planar graphs are so important. This is the dictionary that gave rise to the first string theory. The role of the "string coupling constant" is really as a free parameter. When someone says that string theory has no free parameters means that you can change it by changing the expectation value of a dilaton field. $g_s = e^{\langle \phi \rangle}$. So there are many free choices in the vacuum module and that's where all the free parameters are being hidden. It is a constant only within that module. You might be concerned whether the sum is convergent or only makes sense in $\mathbb{C}[[u]]$, but either way you can still pick out the constant term.

As a source, say look at chapter 3 of Polchinski. You will see it in bosonic string, but that is enough for you to mutatis mutandis and get the A-model topological string.