For everything you're requesting, it seems reasonable only to consider this question using maps that are isomorphisms, since: (i) the zero map between semisimple Lie algebras has no reasonable "associated" map between root systems, (ii) inclusions such as ${\rm{Sp}}_{2n} \rightarrow {\rm{SL}}_{2n}$ have no reasonable associated map either way between the root systems (the latter is simply laced of rank $2n-1$ and the former has two root lengths and rank $n$).
After one restricts the morphisms under considerations to be isomorphisms (which is where all of the substance lies anyway), the answer is "yes". It comes down to a trick, but there is some real content underlying the trick. I have no idea what "strongly monoidal" means, but if one thinks about how mathematicians use semisimple Lie algebras, semisimple Lie groups, and semisimple algebraic groups (after they're done being classified) then such a classification that is functorial with respect to isomorphisms is not only elegant but also useful to do important things.
More to the point, the need for such a version of the classification theorem (really in a form with split connected semisimple groups over general fields in place of semisimple Lie algebras over $\mathbf{C}$) already arose back in the 1960's and especially 1970's, as part of the structure theory of connected semisimple groups over general fields (to classify Galois-twisted forms, especially over fields of arithmetic interest and over $\mathbf{R}$) and in the definition of the Langlands dual group beyond the split case. It is sometimes attributed to Kottwitz, but the main idea can already be found in SGA3 in the discussion of the "scheme of Dynkin diagrams" (see Exp. XXIV, 3.1--3.4) and it has probably been rediscovered multiple times (e.g., by Tits for his notion of $\ast$-action of Galois groups on diagrams when formulating classification theorems over general fields without split hypotheses).
So finally the question comes down to this: is there a way to associate a root system $\Phi(\mathfrak{g})$ to a complex semisimple Lie algebra $\mathfrak{g}$ in a manner that is functorial with respect to isomorphisms? Sure! Let $G = G(\mathfrak{g}) = {\rm{Aut}}_{\mathfrak{g}/\mathbf{C}}^0$ be the identity component of the automorphism scheme of $\mathfrak{g}$. It is a basic fact from the structure theory of connected semisimple groups and semisimple Lie algebras over $\mathbf{C}$ that the groups $G(\mathfrak{g})$ are exactly the connected semisimple algebraic groups with trivial center, and the natural map
$${\rm{Lie}}({\rm{Ad}}_G): {\rm{Lie}}(G) \rightarrow {\rm{Lie}}({\rm{GL}}(\mathfrak{g})) = {\rm{End}}(\mathfrak{g})$$
is an isomorphism onto $\mathfrak{g}$ (embedded via ${\rm{ad}}_{\mathfrak{g}}$). It is also a general fact that the natural action of $G(\mathbf{C})$ on $\mathfrak{g}$ is transitive on the set of pairs $(\mathfrak{h}, \mathfrak{b})$ consisting of Cartan subalgebras $\mathfrak{h} \subset \mathfrak{g}$ and Borel subalgebras $\mathfrak{b} \subset \mathfrak{g}$ containing $\mathfrak{h}$, and that the stabilizer in $G(\mathbf{C})$ of such a pair is the subgroup $T(\mathbf{C})$ where $T \subset G$ is the unique maximal torus whose Lie algebra is $\mathfrak{h}$ (inside $\mathfrak{g}$). For such a pair, let $(\Phi, \Delta)$ be the associated "based root datum" consisting of the root system for $(\mathfrak{g}, \mathfrak{h})$ and the root basis attached to $\mathfrak{b}$ (i.e., the simple roots in the positive system of roots consisting of those whose root lines are contained in $\mathfrak{b}$).
Note that the adjoint action on $T(\mathbf{C})$ on $\mathfrak{h}$ is trivial. Hence, if $(\mathfrak{h}', \mathfrak{b}')$ is another such pair in $\mathfrak{g}$, with $(\Phi', \Delta')$ the associated based root datum, then there is a canonical isomorphism $(\Phi, \Delta) \simeq (\Phi', \Delta')$: the adjoint action on $\mathfrak{g}$ arising from any element $g \in G(\mathbf{C})$ that carries $\mathfrak{h}$ onto $\mathfrak{h}'$ and carries $\mathfrak{b}$ onto $\mathfrak{b}'$ (such $g$ is unique modulo $T(\mathbf{C})$, so this isomorphism between based root data is independent of the choice of such $g$).
We define the canonical based root datum $(\Phi(\mathfrak{g}), \Delta(\mathfrak{g}))$ to be the inverse limit of all such $(\Phi, \Delta)$'s along the canonical isomorphisms just specified. More concretely, elements of $\Phi(\mathfrak{g})$ are exactly the compatible systems of roots with respect to those isomorphisms, and likewise for $\Delta(\mathfrak{g})$. It is clear from the construction that this pair is functorial with respect to isomorphisms among such $\mathfrak{g}$'s. Now compose this functorial construction (functorial with respect to isomorphisms) with the "forgetful" functor that drops the data of the root basis!
Best Answer
Probably the earliest intrinsic definition of Weyl group occurs in section 1.2 of the groundbreaking paper "Representations of Reductive Groups Over Finite Fields" by Deligne and Lusztig (Ann. of Math. 103, 1976, available at JSTOR). This is done elegantly in the closely related but more general setting of a reductive algebraic group $G$ over an arbitrary algebraically closed field (though their interest is mainly in prime characteristic). Letting $X$ denote the set of all Borel subgroups of $G$, the set of $G$-orbits on $X \times X$ provides a natural model for a universal Weyl group of $G$ (or its Lie algebra).
[ADDED] In the algebraic group setting, this intrinsic definition depends just on knowing what a connected reductive (or semisimple) group is and what a Borel subgroup is (maximal closed connected solvable subgroup). But obviously one can't exploit the "Weyl group" without knowing more of the structure theory: conjugacy theorems, Bruhat decomposition. (Is it a group? finite?) In the easier characteristic 0 Lie algebra theory, where $X$ becomes the set of Borel subalgebras (whose definition requires some theory) with conjugation action by the adjoint group, this abstract notion of "Weyl group" similarly needs unpacking. But the Deligne-Lusztig definition is a good conceptual one for their purposes and sneaks in the underlying set $X$ of the flag variety of $G$. Any intrinsic definition of the Weyl group needs serious background in Lie theory.
In the treatment by Chriss and Ginzburg, even when one is primarily interested in the Lie algebra picture, the group in the background tends to play an important role. Indeed, in the early work of Borel and Chevalley on semisimple algebraic groups, the Weyl group appears most naturally in the guise of the finite quotient $W_G(T) :=N_G(T)/T$ for a fixed maximal torus $T$. Then one sees $W$ as generated by reflections relative to roots, etc. As in the parallel Lie algebra setting in characteristic 0, the maximal tori (or Cartan subalgebras) are all conjugate under the adjoint group action, but this falls short of giving an intrinsic definition of the sort provided by Deligne-Lusztig.
[Weyl himself gave the group an awkward name, but was mainly concerned with its use in the context of a compact Lie group. The notion basically originates earlier in the work of Cartan, but it took a while to see the root system and Weyl group as combinatorial objects including the Coxeter presentation of the group as a reflection group (carried over by Witt to Lie algebras).]