a real world point of view
A best boy is the principal assistant of the gaffer and is therefore responsible for the day-to-day maintenance of the electrical department. He co-ordinates and schedules the lighting and electrical work that needs to be done on an episode and manages the personnel issues of his department. Other departments usually communicate to the electrical department through the best boy. Despite its connotations, the best boy is not usually a particularly young person, nor necessarily male.
Best boys were never credited until the 2005 version of Doctor Who. This is probably because, to the extent they existed, they were considered a part of the BBC's general staff, rather than personnel specifically assigned to Doctor Who. Since Rose, however, they have been credited on every episode.
Key grips can also have best boys, but no BBC Wales production has yet credited a best boy grip. When the credit "best boy" is seen in the end credits of a BBC Wales episode, it is refers to the more precise title, "best boy electric".