What do you call this property involving a function between two complete metric spaces?

By continuity, the pre-image of a neighborhood of $b$ should be a neighborhood of $a$. Now if $a$ is a discrete point in $A$ (i.e. the set $\{a\}\subset A$ is open), that neighborhood may be trivial such that the pre-image of the punctured neighborhood of $b$ does not contain any other points in the vicinity of $a$. Otherwise you should be able to find such a sequence. Hence that might be the concept you are looking for.

Let us call the property described in question as Property P. Continuing the observations made in Stefan Böttner's answer we get the following.

Observation. Let $A$ and $B$ be metric spaces and $e\colon A\to B$ be a continuous function. Then $e$ has the Property P if and only if $A$ has no isolated points and $e$ is nowhere constant. (I am not sure to which extent this is a standard therm, but it seems to ba a natural name for this. It also appears in some books.)

By nowhere constant I mean that there is no non-empty open subset $U\subseteq A$ such that $e|_U$ is constant.

Proof. $\boxed{\Rightarrow}$ If $a$ is any point of $A$ then property P implies existence of a sequence converging to $a$, hence $a$ is not isolated.

Let $U\ne\emptyset$ and $a\in U$. Let $b=e(a)$. Let $\varepsilon>0$. The set $e^{-1}[P_\varepsilon(b)]$ contains sequence $(a_n)$ converging to $a$. Starting with some $n_0$, terms of these sequence belong to $U$ and we also have $e(a_n)\ne e(b)$. Therefore $e|_U$

$\boxed{\Leftarrow}$ Let $B(b,\varepsilon)$ be the open ball around $b$. By continuity we get that there is a $\delta$ such that $B(a,\delta)\subseteq e^{-1}[B(b,\varepsilon)]$. Let us choose $n_0$ with $1/n_0<\delta$. Then each ball $B(a,\frac1{n_0+k})$ lies inside $e^{-1}[B(b,\varepsilon)]$. And since the function $e$ is not constant on this ball, we can choose $a_k\in B(a,\frac1{n_0+k})$ such that $e(a_k)\ne e(a)$, i.e., $a_k\in e^{-1}[P_\varepsilon(b)]$.