Suppose $M$ is a monoid and consider an element $x \in M$. Then we call $x$ central iff for all $m \in M$, it holds that $am=ma$. A vast weakening of this condition is to merely require $xM=Mx$. Lets tentatively call this "weakly central." We can equivalently define it as follows: an element $x \in M$ is weakly central iff for all $y \in M$, it holds that $x$ divides $y$ on the left iff $x$ divide $y$ on the right.
This condition shows shows up occasionally in ring theory, see here.
Obviously, being weakly central follows from being central. Interestingly, it also follows from being a unit (so in particular, every element of a group is always weakly central). Anyway, I'd like to know whether there's an accepted way of referring to this condition.
Q. If $M$ is a monoid, is there accepted terminology for those elements $x \in M$ satisfying $xM = Mx$?
I do not think there is any terminology at all for these elements. For an arbitrary monoid and an arbitrary element, it is a pretty loose condition.
For an idempotent $e$, the condition $eM = Me$ implies $eM = Me = eMe$. Here at least the monoid $eMe$ (with $e$ as a unit) occurs frequently in the literature, but I do not know whether the condition $eM = Me$ has been ever considered before.
In the case of a finite (or more generally stable) monoid or in a compact monoid, the condition $eM = Me$ implies that the group of units of $eMe$ coincides with the $\mathcal{J}$-class of $e$, but unfortunately, this latter condition does not suffice to imply that $eM = Me$: take the 5-element monoid $\{1, e, a, ea, 0\}$ with $e^2 = e$ and $ae = 0$. Here $eM = \{e, ea, 0\}$ but $Me = \{e, 0\}$, although the $\mathcal{J}$-class of $e$ is equal to $\{e\}$.