How can I show that if we are restricted to finite structures and take a sentence$\phi$, then for each model $M$ of $\phi$ there is a substructure of $M$, which is a minimal model of $\phi$?
Is this something weaker than the existance of prime model?
Is it not an application of the least number principle? (A variant of the induction principle in arithmetic).
Prime models are different beasts. A model is prime it is elementarily embeddable in every model of a given complete theory $T$. The notion is trivial for finite models. If $M$ is finite, $M$ is the unique model of Th$(M)$ up to isomorphism. (Hence every finite model $M$ is a prime model of Th$(M)$ -- the embedding is actually an isomorphism.)