The names of the characters who ride the school bus with Danny Gopnik are the names of children that the Coen brothers grew up with.
Since minors are only allowed on film sets for only a few hours a day when compared to the adult cast and crew, the sets for Dr. Sussman's bedroom, the Jolly Roger Motel Room, and the sci-fi movie that Danny is watching on the television were all built in the gym of the same high school all the Hebrew school scenes were shot so if any of the scenes involving children were taking too long to shoot, they could knock off any of these other scenes without having to transport the entire crew and adjust the schedule accordingly.
The song heard on the record played repeatedly in the Gopniks' house is Dem Milners Trern ("The Miller's Tears") by Sidor Belarsky, a Yiddish folk song of a sad miller's fears of growing old and alone, echoing the film's theme.
Tyson Bidner, the film's location manager, was cast as the magbiah at Danny's bar mitzvah because he had been one in real life. He said the Torah scroll was very heavy and difficult to lift above his head.
Most of the doorposts throughout the movie (including the Gopniks' and Mrs. Samsky's) have a small box attached to them. This is a mezuzah, a case containing passages from the Torah (Deuteronomy 6:4-9 and 11:13-21), which Jews traditionally affix to the door frames of their houses as a constant reminder of God's presence. A mezuzah also functions as a sign that a Jewish person occupies the house or works in the building onto which it is affixed, so in this movie, the frequent sight of mezuzahs on doorframes is one of many indications that most of the characters are Jewish.