Pick an integral basis $a_i,\ i=1,\ldots,n=b_1(X)$, for $H^1(X,\mathbb Z)$, then form the element $$a:=a_1\cup\ldots\cup a_n\in H^{b_1}(X,\mathbb Z).$$

This is canonical up to sign; any other choice of basis differs by some $g\in GL(H^1(X,\mathbb Z))$, which changes $a$ by $\det g=\pm1$. There is no sign ambiguity on a Kähler manifold because we can choose the complex orientation on $H^1(X)$.

Has anyone seen this thing before ? Does it even have a name ?

It depends on $X$. For instance examples with $b_1(X)=4$ include $T^4$, where it is the volume form, and $\Sigma_2\times S^2$, where it is zero. ($\Sigma_2:=$ genus $2$ Riemann surface.)

I'm not sure what kind of answer I'm after. I have some curve counting invariants on a projective variety $X$ that depend only on the Chern numbers of $X$, and Chern classes evaluated against this strange canonical class $a$. Is that the best I can say ?