Is there a common name for the complement $\widehat{X} \setminus X$ of a metric space $X$ in its metric completion $\widehat{X}$? Since $X$ is not necessarily open in $\widehat{X}$, the term boundary is out of the question (without additional qualifiers). Metric remainder seems appropriate but I did not find it in the literature.

Remainder. I agree with that. But I don't find it online. Maybe "remainder" is primarily used for $\beta X \setminus X$ ? But it should be OK in your setting if you say the first time you use it: "the remainder of $X$ in its completion" or something. 


Corona? Ideal boundary? 


Hausdorff boundary A.P. Kopylov "On unique determination of domains in Euclidean spaces" section 6 "Domains with Hausdorff Boundaries" http://link.springer.com/article/10.1007/s1095800891495. It is posiible to use word "boundary" as part of the name of $\widehat{X}\setminus X$ if $X$ is domain in $\mathbb{R}^n$. 


Penumbra? Cointerior? 

