Can one decide to distract oneself with or by something?

If it's a noun used as an instrumental indirect object, it's "with". Where it would be "by" is if it's causitive ("he was distracted by girl") or an action ("he distracted himself by talking with a girl").

Whether "with" or "by" is often tricky.

In this case, I would analyse it thus:

"He decided to distract himself with a book." In this case, the book is a tool which he is implementing as a technique to distract himself. You do things with a tool.

An example of how you would use "by":

"He was going to church but he was distracted by the boy next door and went fishing instead." In this case, the boy next door is an event which happens outside of his control. You are affected by events outside your control.

The first example you give is another instance of the first of these: the girl is a "tool" which he implements as a distraction technique, however bluntly utilitarian it sounds.

Hence it would be:

"...he decided to distract himself with her friend ..."