"The Doctor has name dropped Harry Houdini so many times now over various incarnations I think we are due a first Doctor companion chronicles where they meet and get caught up in an alien invasion that requires some expert escapism. I think the first instance of his name is in Planet of the Spiders but I am willing to be corrected on that. The Doctor starts to figure how this world works and conjures up doors to escape through. In the world of dreams would a formidable foe; his imagination could probably see a way out of any situation."
Houdini was most recently name checked in The Vampires of Venice, an episode of Matt Smith's first season as the eleventh Doctor. A list of Doctor Who-Houdini references can be found at Tardis Index File.
Thanks to Athena Stamos.