Sherlock Holmes employed a page boy to run errands for him. What was the name of this ambitious young man?

Billy.

Explanation:

The name of the young boy employed by Sherlock Holmes as a page boy was Billy. He plays just a minor character in the Sherlock Holmes series.

The character of Billy can be seen most prominently in the stories "Valley of Fear", "The Problem of Thor Bridge", "The Mazarin Stone" where he even played a significant role in the arrest of the villain. He plays a much more significant character / role in three plays by the author Arthur Conan Doyle. The three plays on Sherlock Holmes are "Sherlock Holmes, A Drama in Four Acts", "The Stonor Case" and "The Crown Diamond". He also appears in "The Painful Predicament of Sherlock Holmes", a spoof about Sherlock Holmes by William Giillette.