Let $X$ be an anabelian curve over a number field $K$ and let $p:Y\rightarrow X$ be a finite etale cover. Then is anything known (or has anything been conjectured) about the field of definition of $Y$? In particular, should it be an abelian extension of $K$?

Certainly not, unless I totally misunderstand your question. For instance, an unramified double cover of X is going to be defined over the field obtained by adjoining a 2torsion point of the Jacobian of X, and that field is going to be nonabelian as all get out. 

