  We classify the nilpotent Lie algebras of real dimension eight and minimal
center that admit a complex structure. Furthermore, for every such nilpotent
Lie algebra $\mathfrak{g}$, we describe the space of complex structures on
$\mathfrak{g}$ up to isomorphism. As an application, the nilpotent Lie algebras
having a non-trivial abelian $J$-invariant ideal are classified up to eight
dimensions.
