It looks so coKleisli, but it's not. What is it? Fix a symmetric monoidal category $(M,\otimes,I)$ and a small discrete monoidal subcategory $M'\subseteq M$. Define a new symmetric monoidal category $C:=CoKl(M,M')$ as follows: $Ob(C):=Ob(M)$, and for any $X,Y\in Ob(C)$,
$$Hom_C(X,Y):=\sum_{m\in M'} Hom_M(X\otimes m,Y).$$
There is thus a projection $Hom_C\to M'$. The identity morphisms project to the unit, $m=I$ and use the identity morphisms from $M$. Given morphisms $\phi\colon X\otimes m_1\to Y$ and $\psi\colon Y\otimes m_2\to Z$ the composition $\psi\circ\phi$ projects to $m_1\otimes m_2$ and is given in the obvious way by 
$$X\otimes (m_1\otimes m_2)\to Y\otimes m_2\to Z.$$ 
The monoidal structure on $C$ is straightforward.
Now doesn't $C$ look like the cokleisli category of some comonad? But of course it's not, in general. There's a functor $M\to C$, but not generally an adjoint. 
If $M=\pi_0(M)$ is a discrete monoidal category, then this construction gives something like the quotient, namely
$\pi_0CoKl(M,M')\cong M/M'.$
Is there a name or reference for this coKleisli-like construction?
 A: I believe that this is a particular case of a colax colimit in $\mathrm{Cat}$, which a coKleisli category is also.  So they are both particular cases of the same construction.
In more detail, a category with a comonad on it is the same as a colax functor $1\to \mathrm{Cat}$.  The input to your construction, which works even if your monoidal functor $S : M'\to M$ is only colax, can similarly be regarded as a colax functor $B M' \to \mathrm{Cat}$, where $B M'$ is the one-object 2-category corresponding to $M'$; the object $m\in M'$ regarded as a morphism in $B M'$ goes to the endofunctor $(-\otimes S m)$ of $C$.
In general, the colax colimit of a colax functor $F:A\to K$, where $A$ and $K$ are 2-categories, is an object $L\in K$ universally equipped with 1-morphisms $q_a : F a \to L$ for objects $a\in A$ and 2-cells $q_a \to q_b \circ F(\alpha)$ for morphisms $\alpha\in A(a,b)$ that respect all the structure of $F$.  The fact that the colax colimit of a comonad in $\mathrm{Cat}$ is its coKleisli category, or (as more commonly stated) that the lax colimit of a monad is its Kleisli category, dates back to Ross Street's formal theory of monads, 1972.  I believe that the same sort of argument should show that your construction is the colax colimit of the functor $B M' \to \mathrm{Cat}$.
A: 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 $M'\to M$.
A: This is close to the notion of orbit category appearing for example in


*

*Bernhard Keller, On triangulated orbit categories, http://arxiv.org/abs/math/0503240
and section 7 of


*

*Gonçalo Tabuada, Chow motives versus noncommutative motives, http://arxiv.org/abs/1103.0200.pdf
The orbit category of an additive category $A$ with respect to an auto-equivalence $T : A \to A$ is the category whose objects are the same and whose morphism groups are given by
  $$ \mathrm{Hom}(X, Y) = \bigoplus_{i \in \mathbf{Z}} \mathrm{Hom}_A(X, F^i(Y)). $$
It is typically denoted $A/T$.
The second reference discusses the symmetric monoidal case and will be more interesting for you.
Since you mentioned something about it (though I am not sure I understand what you wrote), let me also mention that the projection functor $\pi : A \to A/T$ admits a right adjoint $\tau : A/T \to A$ which sends an object $X$ to the sum of all the $F^i(X)$ (assuming that $A$ admits infinite direct sums).
