I love this word. It connotes dogged effort in obtaining something: specifically, a patient working at something with one's fingers, wiggling it free. Think of a bent nail in a board. What do you do? Nag at it, work at it. Or better yet: think of a bit of meat between one's teeth. One keeps at it with tongue and tooth until it comes loose. We do not use finagle with these situations, because finagle refers only to a metaphorical wiggling loose. Did the word in the past have a "physical" meaning? I do not know, but I suspect so.
Now it is mostly used in bureaucratic situations, to convey the effort and flexibility needed to wiggle a desired result out of a complex process. "Let's try to finagle it so we are approved for a business rather than a personal loan." Actually, there is an element of manipulation in this example: the weak side using strategy and opportunism to get a certain result.
Is this word yet another Irish Gaelic addition to english? (see article below). No, what am I thinking? This word must be from Yiddish. It rhymes with "bagel," right? Being a linguist is really so easy.
Today at the place i work an old woman poked her head in the kitchen and asked, "could I finagle a cup of milk?" In using the word, she was exaggerating to humorous effect the labor and difficulty of her own request.