It may be small, but the convict baseball is a pretty important object in this story. It helps bring a lot of different things that are going on together, plus it adds an extra dose of drama to the tale. So let's take a look.
The baseball's appeal comes from the kids' obsessions with all things convict related. Other than that, it is just an ordinary baseball, nothing fancy about it—as Moose tells Scout: "They look just like any other baseball" (23.19). The baseball, then, is a reminder that these kids turn ordinary things (like baseballs and getting their shirts washed) into Big Deals just because of the idea they're connected to—a.k.a. the convicts.
But there's another reason that the ball is a big deal for Moose: competition. If he finds a ball for Scout before Piper does, he wins. And Moose becomes obsessed with finding the ball. He even dreams about it: "Last night I dreamt that Scout found the gap. The thought of Scout worming his way through my hold in my fence and finding my ball makes me nuts" (25.3). Finding the ball first becomes tied to Moose's sense of self—it's a way of laying claim to both his new home on Alcatraz, and his new friend (a.k.a. Scout) at school.
Now how's this for a series of events: If Moose wasn't so worried about finding the baseball, Natalie wouldn't run into 105. In a twist, though, if Natalie hadn't met 105, Moose never would have found a baseball:
"You want this?" He reaches inside the coat draped over his leg. […]
It's a baseball. (26.8-9)
What matters most here is that Natalie is key to Moose finally getting the ball. Throughout the book, she's presented as a sort of liability—someone who needs to be watched after, no matter what, and who kind of gets in the way. Here, though, she's essential to Moose's success. And because the baseball is tied to Moose coming into his own in this new chapter of his life, and Natalie helps him get the baseball, she acts like a classic older sibling, showing that despite her differences or difficulties, she's still Moose's big sister.
But as with all things connected to Natalie in this story, this isn't the whole picture: In helping Moose get the ball he so desperately wants, the baseball also becomes a sign of Moose's foolish behavior. He says: "I feel like I held my sister hostage for that stupid baseball. I won't touch it. It's dirty" (27.21). Moose feels super ashamed that he was so worried about keeping up with the kids at school that he let his sister befriend a convict. While finding the ball is evidence of Natalie's role as his big sister, in his guilt, we're also reminded that in many ways, Moose is the older sibling, even though he's technically younger.