How do you procedurally create a mesh that appears to thickly drip (or elongate to appear to drip) from the point of impact on another object that it appears to "hang off"?
There might be other ways to create this effect, but I think the question boils down to: Basically, how do you smoothly elongate a "point" on a mesh in a certain direction?
It's a non-trivial question, but to answer the specific question of how to animate a single vertex of a mesh, you'll want to start here, and perhaps look at the 'procedural mesh' examples in the 'resources' section of the Unity site as well.
Once you've identified the vertex that you want to animate, updating its position can be done in more or less the same way you'd animate anything else. (For example, you could simply adjust the element of the vertex corresponding to the vertical direction according to a 'drip speed' and the current time step on each update.)
Note that this is unlikely to produce a particularly realistic effect, as animating a single vertex in this way is likely to produce a noticeable discontinuity in the mesh (rather than a smooth, continuous deformation).