I don't know what Hilbert actually intended with that quote as I haven't read the context let alone exhaustively gone through Hilbert's writings with this quote in mind. I strongly suspect Hilbert meant this in a rather informal way. Nevertheless, let's brazenly charge forward with trying to take this fairly literally. As a nice touch, what I discuss is quite friendly with Formalism.
One interpretation is the mathematicians are (or should be) focused on discovering and analyzing things like free objects, e.g. the free monoid. For a class of relevant properties, any of those properties that hold of the free object hold for every other similar sort of object. These free objects are often rather syntactical. (A significant generalization of this is the notion of syntactic categories and internal languages. A different object that has similar properties is classifying spaces/classifying toposes.)
Continuing the spirit of being overly literal, a good example of this is the Lindenbaum-Tarksi algebra for concreteness say for classical propositional logic. This is essentially a free Boolean algebra. It has the property that a formula of classical propositional logic is valid if and only if it is true in this particular model. This is a very literal interpretation of a "special case" that contains the "germs of generality".
Being less literal but maybe closer to the spirit of Hilbert's statement. Syntax is the handle we have on mathematical objects. To the extent that we want to actually calculate things (often even for the very amorphous notion of "calculate" that mathematicians often use), we need a (more) syntactic/combinatorial description of an object. Often that isn't quite possible, but what is possible is finding an equivalent object that is describable in a more combinatorial manner. Probably one of the best examples of this is the notion of a simplicial complex from algebraic topology1. I'm fairly confident that Hilbert would view simplicial complexes as an example of a special case containing the germ of generality.
1 To connect to earlier, the closely related notion of a simplicial set gives rise to the category of simplicial sets and is a classifying topos.