In my paper Disjoint $n$-amalgamation and pseudofinite countably categorical theories, I did some work on this question for the restricted class of countably categorical theories. (The question is already very hard for these theories! For example, the finite model property for the theory of the generic triangle-free graph is a famous open problem.)

A theory $T$ has disjoint $n$-amalgamation if, for any system of complete types $\{p_X((x_i)_{i\in X})\mid X\subsetneq [n]\}$ which agree on their intersections, there is a type $p_{[n]}((x_i)_{i\in [n]})$ extending all of them. For example, disjoint $3$-amalgamation means that whenever three $2$-types $p_{\{0,1\}}(x_0,x_1)$, $p_{\{0,2\}}(x_0,x_2)$, and $p_{\{1,2\}}(x_1,x_2)$ agree on their intersection $1$-types $p_0(x_0)$, $p_1(x_1)$, and $p_2(x_2)$, then there is a $3$-type $p_{\{0,1,2\}}(x_0,x_1,x_2)$ extending all three. This condition fails for the generic triangle-free graph, since you can't amalgamate three edges into a triangle.

I observed that if $T$ is countably categorical and has disjoint $n$-amalgamation for all $n$, then $T$ has the finite model property. The proof is by a probabilistic argument very similar to the usual proof of the finite model property for the theory of the random graph. This sufficient condition generalizes other sufficient conditions identified previously (see p. 9 of the linked paper), such as the *parametric universal theories* (Oberschelp), and *local Fraïssé classes* (Brooke-Taylor & Testa).

In my view, the really interesting thing is that all of the countably categorical theories which I know to have the finite model property have either a "structured/algebraic" flavor (think infinite-dimensional vector spaces over a finite field) or a "random/combinatorial" flavor (think the random graph). And every "combinatorial" example that I'm aware of can be explained by disjoint $n$-amalgamation, either directly (as in the random graph, for example) or indirectly by approximating the theory and taking finite expansions of the language (as in the theories $T^*_{feq}$ and $T_{CPZ}$ analyzed in the linked paper).

It would be very interesting to somehow make these ideas precise, but I haven't yet seen how to do it (again, it seems very hard: a successful realization of this goal would show that the theory of the generic triangle-free graph does *not* have the finite model property, since it is definitely "combinatorial", but it has a very strong obstruction to $3$-amalgamation). So instead I'll just leave you with two quotes:

“In all those homogeneous structures which I know to have the finite
model property, [it] arises either from probabilistic arguments as
above [i.e. 0-1 laws], or from stability, or conceivably from a mixture of
these.”
- Macpherson, *A survey of homogeneous structures*

“When does a homogeneous structure for a finite relational language
have the finite model property? More broadly, is there anything of
interest in graph theory besides randomness and algebra?”
- Cherlin, Exercises for logicians (worth a read, if you haven't seen them before!)

sufficientconditions, but in some sense explains the complicatedness of this topic. $\endgroup$ – Peter Heinig May 17 '17 at 15:393more comments