Why do counits go that way? Imagine you want to motivate for an audience the definition of an adjunction in terms of unit and counit. So you can say: Often two functors
$\mathcal{C} \begin{array}{c} \stackrel{\large F}{\rightarrow} \\ \stackrel{\leftarrow}{G}\end{array} \mathcal{D}$ are not really inverse, up to isomorphism, but still, the compositions $F \circ G$ and $G \circ F$ are somehow "connected" to the identity functors. So we have natural transformations
$$\eta : \mathrm{id}_{\mathcal{C}} \to G \circ F,$$
$$\varepsilon : F \circ G \to \mathrm{id}_{\mathcal{D}},$$
which should be compatible in a suitable way. Then someone says:
"Wait, why not taking $\varepsilon : \mathrm{id}_{\mathcal{D}} \to F \circ G$ instead? What's wrong with that?"
What would you answer?
Just to be clear: I understand the concept of an adjunction and know that $\varepsilon$ has to go from $F \circ G$ to $\mathrm{id}_{\mathcal{D}}$, but I am not sure how to explain this  to a novice without going into the details of the notion of an adjunction, for example the triangle identities or the alternative hom-set-definition, which by the way I would like to exclude from the discussion here. A good answer should be intuitive.
Notice that it is not true that we cannot state any compatibility conditions between two natural transformations $\eta : \mathrm{id}_{\mathcal{C}} \to G \circ F$ and $\varepsilon : \mathrm{id}_{\mathcal{D}} \to F \circ G$. An obvious candidate would be
$$F \circ \eta = \varepsilon \circ  F : F \to F \circ G \circ F$$
and/or
$$\eta \circ  G = G \circ  \varepsilon : G \to G \circ F \circ G.$$
By the way, does this concept have a name?
 A: The unit-counit definition of an adjunction makes sense in any 2-category, and so it makes sense in any monoidal category, where it recovers the definition of a dualizable object. So you can motivate this somewhat more general notion using the familiar special case of vector spaces: when a vector space $V$ has a dual $V^{\ast}$, there is a unit map $1 \to V^{\ast} \otimes V$, namely the coevaluation, and also a counit map $V \otimes V^{\ast} \to 1$, namely the evaluation. 
Of course it's hard to distinguish between the two possible orders of the tensor product here since $\otimes$ is symmetric monoidal. At this point I would draw the triangle identities using string diagrams, which justifies using the orderings above, although if you're going to both avoid that and also the homset definition of an adjunction I don't know what to tell you. 
A: Maybe this veers too close to talking about the triangle equations, but I find it suggestive that it's possible to paste the unit and the counit in some way when the counit goes the right way around, whereas if it goes the wrong way around, there's simply no pasting that you can possibly do. With no pasting, there wouldn't be anything you could "do" with the data of an adjunction, except maybe to have it interact with other adjunctions.
Maybe you could make something of the following fact: if a functor is a self-adjoint endofunctor, and you have the counit the wrong way around, then the unit and counit have the same domain and codomain. Maybe that seems a little weird.
Otherwise -- examples?
EDIT
From the way the question is worded, it sounds like you want to motivate the adjunction concept as a weakening of the equivalence concept. Perhaps the question "why doesn't $\epsilon$ go the other way?" comes from the fact that when we're talking about equivalences, it often is presented as going the other way. Then it will make sense to point out that in the equivalence case, since the 2-cells are invertible, there really are 2-cells going in both directions, and so we're free to generalize it by choosing whatever direction we please. Then you can motivate the actual choice of directions by saying "wait and see the cool equations that this allows me to impose".
A: tcamps gave what I think is the correct answer (nice pasting diagrams justify the choice formally, examples justify it's usefulness), but here's an alternate approach which might have some pedagogical merit, if you don't think it's too roundabout.


*

*Use the adjunction isomorphism as the definition, or state that it's equivalent to the (unit, counit) one you are presenting. The latter is better because it generalizes, but the adjunction isomorphism I think is easier to motivate, because there's couple of really obvious examples (familiar examples of free algebras, e.g. free modules, and familiar examples of monoidal closure, e.g. Set).

*Only thing you know for sure a general category has are identities. The two most obvious things you get using identities and the adjunction isomorphism go in different directions.
A: Firstly, as tcamp also commented, there also exists a similar construction as adjunctions in any bicategory where the 'unit' and 'counit' go in the same direction, and these are the (dual) Morita contexts, and it is also worth to study, having already wide literature, at least in its origin world of rings. 
(Note that adjoint equivalences are immediately Morita [and dual Morita] contexts by flipping the [co-]unit.)
Secondly, consider the two (pseudo) embeddings of bicategories ${\bf Cat}\to{\bf Prof}$ sending $F\mapsto F_*$ and $F\mapsto F^*$, where $F_*=(a,b)\mapsto\hom_B(Fa,b)$ and $F^*=(b,a)\mapsto\hom_B(b,Fa)$ for a functor $F:A\to B$.
Observe that both of these are contravariant in exactly one dimension ($F\mapsto F^*$ on arrows and $F\mapsto F_*$ on 2-cells).
Moreover, we can roll these two maps in one embedding of double categories ${\bf Q}({\bf Cat})\to{\bf Q}({\bf Prof})$ which is contravariant in exactly one direction, where ${\bf Q}(B)$ means the double category of Ehresmann quintets of the bicategory $B$. 
Consequently, orthogonal adjoint pairs in ${\bf Q}({\bf Cat})\ $ (i.e. adjoint functors with counit-unit-definition) correspond to companion pairs in ${\bf Q}({\bf Prof})\ $ (which yields the "$L_*\simeq R^*$" definition).
A: I am not sure one can give a justification from scratch. However, an important algebraic motivation of adjunction is to formalize the "free construction" on a set in a given algebraic theory. Leaning on the basic free constructions (groups, abelian groups, modules, rings, etc.) that your student should know, you have a panel of adjunctions $F \dashv U$ to show off, where $U$ is some kind of "underlying set" functor.
Now, the counit can only go in the direction $FU \to \mathrm{id}$ : $FU(x)$ is the algebraic object of "formal combinations" of elements of $x$ ; one can certainly transform a formal combination into a concrete computation, but the other way around would be highly non canonical.
