What might a pediatric neurologist be looking for when looking at a child's privates?

Syndromes. Part of the exam for abnormalities in children is looking and sometimes measuring the size of the testes and penis. Some syndromes have findings which would correlate with this.
Puberty staging. Examination of a child's genitals is how we assess if puberty has started. Depending on the age this can be a good or bad thing. Severely brain damaged children often start puberty very early and may need medication to help slow it down. Also, depending on the issues, looking to see if there is abnormal genitals that might clue into a diagnosis. A complete medical exam always includes the genitals.