Associating a particle with a classical field is what quantization does, practically by definition. Take a classical field, make it an operator, and find the eigenstates of its Hamiltonian. The result is particle states, whatever form the field takes. "Graviton" is just the name we give those hypothetical particles.
Even though it's basically just a placeholder name, we can deduce a few properties the graviton ought to have from the general features of quantization. The metric tensor of general relativity is a rank-2 tensor which tells us immediately that the field quantum would have to be a spin-2 particle. The argument is the same as that which tells us that a scalar field has spin-0 particles associated with it, a spinor field has spin-1/2 particles, etc., and is completely general with respect to the form of the field. Also, the metric tensor has a gauge symmetry, which would make the graviton a gauge boson like the photon. Finally, gravity appears to propagate at the speed of light, and hence a gauge boson mediating it would have to be massless.