We probably wouldn’t ask what makes  graph **properties** useful. In many ways we consider isomorphic graphs as “the same.” Invariants are just properties that respect this sameness.  The specific vertex set is not an invariant. The number of vertices is. You can certainly make up unmotivated invariants like “the number of vertices whose degree is a divisor of $65$”

If you want to decide  “is graph $G$ isomorphic to graph $H$?” then easily computed invariants like number of vertices might easily tell you no. If they fail then you can try harder. But invariants are useful for more than deciding isomorphism.

The girth (length of the shortest cycle), chromatic number, clique number all seem pretty useful. A canonical labeling won’t get you very far toward determining what they are.

As far as how one would create a “good” graph invariant, I think that isn’t the right thing to ask. Instead, start with a question you find interesting and see what invariants it leads to.

You might start with a  question like “when can a graph be drawn in the plane without edge crossings?” Which is itself an attractive invariant. Then you could be drawn to thinking “well, all but one with up to 5 vertices...” and end up with the useful but not obvious idea of graph minors which turn out to be widely useful.