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).