What did Mr. Burke give to Jesse that belonged to Leslie?

This is an important question to consider. After Leslie's passing, or anyone's passing, divvying up that person's belongings can be a very meaningful and cathartic activity. We can pay special attention to scenes like these and understand more deeply the bonds that existed between the person who died and those who meant the most to her.

Let's look in Chapter 13, called "Building the Bridge." Jess is talking with Bill and Judy (the Burkes, who are Leslie's parents.) Here's what he has received so far:

They gave Jesse all of Leslie's books and her paint set with three pads of real watercolor paper. "She would want you to have them," Bill said.

Then, Jess gets one more thing right before the Burkes leave:

"Well," Bill said. "If there's anything we've left, that you want, please help yourself."

"Could I have some of the lumber on the back porch?" Jess asked.

"Yes, of course. Anything you see." Bill hesitated, then continued. "I meant to give you P. T.," he said. "But" - he looked at Jess and his eyes were those of a pleading little boy - "but I can't seem to give him up."

"It's OK. Leslie would want you to keep him."

So we've learned that Jess got Leslie's books, her paint set, her pads of watercolor paper, and some lumber from her back porch. But not P. T., the beloved dog. 

Although the books and art supplies make sense as a gift of remembrance of Leslie, that lumber deserves a bit of consideration. Why would Jess want it? How will he use it to honor Leslie's memory? If we read a bit further, and we if think about the title of the chapter, too, we see that he uses the lumber to build a bridge back to Terabithia. Then he invites his sister May Belle to play with him there. The lumber is a significant gift, then, because Jess uses it to keep on doing what Leslie inspired him to do: be creative, play, be imaginative, find the joy in life, and connect with other people.