When Joe O'Malley and Patrick O'Malley are sitting together in the backyard, we see Patrick holding a beer. At the end of the scene, Patrick has no beer, and Joe is "ceremonially" handing him his first beer.
In the second swimming scene, Pete and Danny's swimming suits each change color. Pete's start out as blue, then become red, then switch back to blue, then back to red. Danny's start out as blue then change to green later in the scene.
When Pete O'Malley and Danny Jacobsen are in Pete's room after Pete gets into trouble, Danny's hat repeatedly changes position between shots.
When Mrs. O'Malley drives up to church with all the kids, after the car is parked and all the kids are out, a head pops up in the back seat and someone opens up the back door. On the DVD director's narrative, it is explained that this is the mother of one of the child actors who had been crouching down during the scene.
The film is set in 1976, but during one of the beach scenes, we see several buildings on the Chicago skyline that were not built until several years later (e.g., the new Prudential Building "Pru-2").
Pete is playing with a Mattel Football handheld game which wasn't invented until 1977, but the movie is set in 1976.
Although set in 1976, modern cars and trucks can be seen parked on the street in the background when Margaret is driving the car with her children at the beginning of the film.
Although the film is set in the mid-1970s, the ambulance at the scene of the house fire is a late-80s model.