Edit. After writing the answer below I realized it is essentially the same as the one by David Lehavi. As the different phrasing may improve understanding I will, reluctantly, not erase it.
If we consider the Poincaré metric (constant curvature $-1$) on an hyperbolic curve all the points are locally the same. I am afraid this implies that there is no local characterization of Weierstrass points. Of course this does not exclude the geodesic focusing property Matt suggests.
I like to think on Weierstrass points as a concept of projective differential geometry. It is an extension of the concept of inflection point of a plane curve.
Indeed if $C$ is a smooth plane quartic then its Weierstrass points are exactly its inflection points, i.e., points at which the tangent line has abnormal order of contact with the curve.
For a curve with very ample canonical bundle( the generic case when $g\ge 3$ ) essentially the same interpretation holds. More precisely we have that
- the canonical bundle $K_C$ determines an embedding of $C$ into the projective space $\mathbb P H^0(C,K_C)^{\ast}$;
- the Weierstrass points correspond to the points in $\mathbb P H^0(C,K_C)^{\ast}$ at which the image of $C$ and the corresponding osculating hyperplane have abnormal order of contact.
