This is close to the subject of my (so far unfinished) thesis, so I'll try to explain for the benefit of future readers.
We define an oplax action of a monoidal category $\mathcal X$ upon a category $\mathcal C$ to be an oplax monoidal functor $\mathcal X \to \text{End}(\mathcal C)$; i.e., a functor $\_.\_\colon \mathcal X \times \mathcal C \to \mathcal C$ together with natural transformations $(x\otimes y).a \to x.y.a$ and $I.a \to a$ satisfying some coherences.
It is easy enough to check that if $\mathcal X$ is the unit category, then this is just the same thing as a comonad on $\mathcal C$. Specializing in another direction, if $\mathcal C$ is also a monoidal category, then any oplax monoidal functor $j\colon \mathcal X \to \mathcal C$ gives rise to an oplax action of $\mathcal X$ upon $\mathcal C$ via
$$
x.a = j(x)\otimes a\,,
$$
in much the same way that if $b$ is a comonad in a monoidal category $\mathcal C$, then there is a monad on $\mathcal C$ given by $Ta = b\otimes a$ (when $\mathcal C$ is Cartesian, every object has a unique comonad structure given by the diagonal and this is called the reader monad).
It would be nice to generalize the co-Kleisli category to arbitrary lax actions, and this is easy to do. The category we get has the same objects as $\mathcal C$, but the set of morphisms $a$ to $b$ is given by the colimit
$$
\varinjlim_{x\colon\mathcal X} \mathcal C(x.a,b)\,,
$$
where the composition of $f\colon a \to x.b$ with $g \colon b \to y.c$ is given by the composite
$$
a \xrightarrow{f} x.b \xrightarrow{x.g} x.y.c \to (x\otimes y).c\,.
$$
If $\mathcal X$ is discrete, then the colimit above is of course a sum.
Why is this a natural generalization of the Kleisli category? One answer is Mike's above: just as the co-Kleisli category of a monad is its oplax colimit when the monad is considered as a lax functor $1 \to \mathbf{Cat}$. Similarly, the category we have described above is the oplax colimit of the action, when we consider it as a functor $\mathbf{B}\mathcal X \to \mathbf{Cat}$.
In fact, there is a nice recipe for constructing arbitrary oplax colimits of lax functors in $\mathbf{Cat}$, which is set out nicely on the nLab. We start by taking the Grothendieck construction of the lax functor, which gives us a bicategory, and then we turn the bicategory into a category by identifying $1$-cells that are related by a $2$-cell.
In the particular case of a lax functor $\mathbf{B}\mathcal X \to \mathbf{Cat}$, corresponding to an action of the monoidal category $\mathcal X$ upon a category $\mathcal C$, the bicategory we end up when we take the Grothendieck construction has objects given by the objects of $\mathcal C$, $1$-cells from $a$ to $b$ given by morphisms $a \to x.b$ in $\mathcal C$ and $2$ cells from $f\colon a \to x.b$ to $g\colon a\to y.b$ given by morphisms $h\colon x\to y$ in $\mathcal X$ such that $f;(h.b)=g$. To turn this bicategory into a category, we identify all $f,g$ that are related by such an $h$, which gives us the same thing as the colimit formula above.
Another way to think of the Kleisli category is that it is what we get when we embed functors into profunctors. A comonad on a category $\mathcal C$ is a comonoid in the category of endofunctors on $\mathcal C$, which means that it gives us a monoid in the category of endoprofunctors on $\mathcal C$. But a little thought (see here, for example) tells us that a monoid in the category of endoprofunctors on $\mathcal C$ is the same thing as a category that admits an identity-on-objects functor out of $\mathcal C$; when we start with a comonad, the category we get is that comonad's co-Kleisli category.
If instead of a monoid in the category of endoprofunctors on $\mathcal C$, we consider a lax monoidal functor out of a monoidal category $\mathcal X$, we see that the structure we get is that of an $[\mathcal X,\mathbf{Set}]$-enriched category, where $[\mathcal X,\mathbf{Set}]$ carries the Day convolution product. Our construction can also be understood in this guise. Given an oplax action of a monoidal category $\mathcal X$ on a category $\mathcal C$, we may define an $[\mathcal X,\mathbf{Set}]$-enriched category on the object set of $\mathcal C$ where the object of morphisms from $a$ to $b$ is the functor
$$
x \mapsto \mathcal C(x.a,b)\,,
$$
composed in the Kleisli style. If $\mathcal X$ is small, we can get back to our category by change of base along the colimit functor $[\mathcal X,\mathbf{Set}]\to \mathbf{Set}$.
I am not quite sure what to call it. There is a reference to the $[\mathcal X,\mathbf{Set}]$-enriched category, in some unpublished notes of Melliès (see part 3), so the $[\mathcal X,\mathbf{Set}]$-enriched version could be called the Melliès category. Otherwise, I think your idea to call it the 'quotient' is good, or even just to call it the 'co-Kleisli category' associated to the action that you get from your inclusion $M'\to M$.
