I believe the historical motivation for considering the spin representation is actually mathematical physics. In his attempt to formulate a counterpart of the Schrodinger equation in quantum mechanics which is compatible with special relativity, Dirac decided that he needed to find a first order differential operator whose square is the Laplacian:
$$ D = \sum c_i \partial_i $$ $$ D^2 = -\sum \partial_i^2 $$
If you sit down and work out the relations that the $c_i$'s have to satisfy, you find that $c_i c_j + c_j c_i = 0$ if $i \neq j$ and $c_i^2 = -1$. Of course, there are no real or complex numbers which satisfy these relations; Dirac realized (not in this language) that the $c_i$'s are generators for the Clifford algebra $\mathbb{C}_n$. Thus $D$ is a vector valued operator which takes values in a vector space $S$ on which $\mathbb{C}_n$ acts by endomorphisms. Ideally we would like to take $S$ as small as possible, in the sense that $End(S) \cong \mathbb{C}_n$.
If this physicsy motivation isn't enough, Atiyah and Singer discovered a few decades later that Dirac's idea has profound applications in differential topology. One says that a Riemannian $n$-manifold $M$ is spin if there is a principal $Spin(n)$-bundle $P \to M$ equipped with a bundle map $P \to SO(TM)$ (where $SO(TM)$ is the principal $SO(n)$-bundle of orthonormal frames) which on fibers is the double cover $Spin(n) \to SO(n)$. If $P$ exists then there is a natural vector bundle $S \to M$ coming from the spin representation via the "associated bundle" construction, and $S$ has the property that its endomorphism bundle is isomorphic to the Clifford algebra bundle associated with $TM$. In this case there is a global "Dirac operator" $D$ whose square is the Laplacian, and this operator is central to the Atiyah-Singer index theorem. In fact, it serves as the fundamental class in real K-homology theory.