When Henry's mother (Amy Morton) is celebrating Henry scoring a run after being walked, she hits her head on the lamp hanging above her (at around 57 mins). Morton obviously didn't mean to do this: she is visibly hurt and after sitting down mutters "oh... shit" (at 57:46, which is dubbed out on the audio, you have to lip-read).
When Henry first goes to Wrigley Field as a pitcher, he goes to the players entrance. When he knocks on the door to be let in, an old man pokes his head through a hole in the door. At first he doesn't let Henry in, then Henry reveals who he is, and the old man says, "Well that's a horse of a different color," which is the same thing that was said in The Wizard of Oz (1939) when Dorothy and the gang get to Oz.
When Henry has to take his first AB (against the L.A. Dodgers), the pitcher in that scene (Tregoraw) is Tim Stoddard. Stoddard was a technical adviser for the baseball scenes in the movie. Coincidentally, Stoddard was actually a pitcher for the Cubs in real life, he pitched on the '84 Cubs team that won the NL East title (and eventually lost to the San Diego Padres in the NLCS).
Filming for the Little League scenes took place at the Little League fields at the park district in Westmont, Illinois, which is a suburb of Chicago. Some locals were used as extras in the stands on the field.