"Painting" Along A 3D Object Radially From A User-Selected Point

Okay, so I have this idea for a puzzle. In it, the player will be able to apply a daub of something, let's say paint, to an object. And the paint moves out from that point a certain amount. They can only apply X number of daubs, and they have to cover the object completely with their limited number of daubs.

If I wanted to represent something like this in Unity, how in the world would I go about that? The object is going to be an actual 3D object, so let's say a simple cube for demonstrative purposes. I want the player to be able to select an area on the cube (probably not difficult--"on mouse over" show a black dot in center of screen, and then on mouse click paint the dot or something on the object, or get the raytraced collision point on the object), and then have something radiate outward (on the 3D object) from where they selected.

I don't have slightest clue how I would do anything like this (the second part mainly, the thread title), and only barely have any idea how to look for something like this. Can anyone give me any tips on where to begin?

 

Anything? Anyone?