What would be a good introductory paragraph for an essay about Bud not Buddy?

I am going to assume that the essay is not a basic book summary.  The first thing to figure out for the entire essay is a thesis statement.  What is your essay going to argue, and what is it trying to prove?  A thesis statement cannot be a statement of fact, so be sure to come up with a thesis that is debatable.  Once you have a thesis figured out, it will guide the entire paper as well as the introductory paragraph.  I would come up with a thesis that focuses on a theme or a character's development over the course of the novel.  One example could be the following:
"Although Bud has gained his real grandfather by the story's conclusion, it is the people that Bud met along the way that are likely to have the longest lasting impacts on his life."  
If this is your thesis, the opening paragraph should briefly focus on some of the most important people that Bud met over the course of the story.  It is up to you to choose who you feel is most valuable, but I would seriously consider Lefty Lewis, Deza Malone, Steady Eddie, and Bugs. 
There is one other part of an introductory paragraph that needs to be given a lot of attention: the first sentence.  I call it the "attention getter."  It needs to grab your reader's attention right away and motivate him or her to keep reading.  A reader that is bored from the first sentence is not motivated to keep reading.  
Asking the reader a question is always an effective way to get the reader's attention.  It forces the reader to think about the question and come up with possible answers.  The reader will also likely be curious about what your answer might be.  
Another effective attention getter is to use a quote.  This could easily be done with this book because Bud is full of great one-liners and rules for life.  You could probably pick any of his rules and relate it to the thesis about the importance of family and relationships to Bud. 

Rules and Things Number 29: When you wake up and don't know for sure where you're at and there's a bunch of people standing around you, it's best to pretend you're still asleep until you can figure out what's going on and what you should do.