Conditions for which a topological space $X/A \cong X$, where $A \subsetneq \partial X$. Say $X \subseteq \mathbb R^n$ has a sufficiently nice structure such as being a compact manifold with boundary  (I'm more interested in standard spaces than anything else.)
Is it true that if $A \subsetneq\partial X$ is a closed and contractible proper subset of $\partial X$, then $X / A$ is homeomorphic to $X$?
It's clear that the homology agrees by excision, but I'm not sure if its easy in general to see such a homeomorphism. I think that in low dimensions that such a result would be reasonable, but I'm not sure how one would prove such a statement.
I suppose  that $2$-manifolds (with boundary) will not change genus, so we can expect this result for manifolds in three dimensions. $1$- manifolds will not change either, which I think is clear.
 A: The answer is negative in dimensions $\ge 4$. The relevant construction is due to Bing: Start with a wild arc $A$ in $S^3$ such that $\pi_1(S^3-A)\ne 1$. Take the quotient $Y:=S^3/A$. It is not a manifold. Indeed, denoting $a$ the projection of $A$ to $Y$, we notice that $a$ does not have a basis of neighborhoods $U_i$ with $\pi_1(U_i -a)=1$.  To obtain an example you are interested in, consider $X=B^4$. 
On the other hand, if you take a surface $S$ (maybe with boundary) and contract a compact contractible subset $A\subset S$ to a point, then the quotient is still a surface (this is due to Moore, who proved much more) and hence homeomorphic to $S$. (It suffices to know that $A$ is "cell-like"; you can also take the quotient of $S$ by an upper semicontinuous partition of it into cell-like compact subsets.)   
From this it follows that your question has positive answer for $n=1, 2$ and, most likely, $3$ (but the latter requires a bit more thought).   
Edit 1: see also this question and references given there. 
Edit 2. The statement indeed holds in dimension 3 as well.
Theorem. Suppose $M$ is a 3-dimensional manifold with boundary, $A\subset \partial M$ is a compact contractible subset. Then the quotient space $M/A$ is a manifold with boundary, map $M\to M/A$ is a homotopy equivalence which is homotopic to a homeomorphism. 
The proof uses mostly the Dehn Lemma.  
