Presumably named after Brockton, MA, since it’s the most prominent Brockton around, but the why remains a mystery. I can tell you that it was originally called 16th Street when Sawtelle was its own city. In 1919, during L.A.’s protracted annexation of Sawtelle, twelve streets were renamed in honor of Civil War battlefields – see Corinth Avenue for the full list – and 16th Street became Jonesboro Avenue, for the 1864 Battle of Jonesboro in Clayton County, GA. (Historians claim it was spelled “Jonesborough”, but 1860s newspapers spelled it “Jonesboro”.) Property owners on the street soon had its name changed to Brockton. Their argument wasn’t publicized – perhaps a homesick Massachusetts transplant suggested it; perhaps someone just hated the name Jones.