"Archie Leach" is the birth name of Cary Grant, star of To Catch a Thief (1955), as well as the lead character played by John Cleese in A Fish Called Wanda (1988). Both films, like most "Leverage" episodes, involve heists.
When Hardison and Sophie enter the building they use the aliases "Emily Peel" and "Jonathan Steed". Emma Peel and John Steed were the main characters in the TV series and movie versions of The Avengers (1961), The Avengers (1998)
At the severance party, Brian says to his sexting buddy (who he believes to be Parker), "I have been, and forever shall be, your Rocket Man." He is referencing a famous line from Star Trek II: The Wrath of Khan (1982), which Hardison also quotes in the season 2 episode "The Three Strikes Job."
Actress Lisa Brenner, who plays Dr. Hannity, is married to producer/director Dean Devlin.
Archie says that he found Parker when she tried to lift his wallet as a child, and took her under his wing to train her. This scenario is similar to the origin story of Storm from the Marvel Comics series X-Men. Jim Steranko, after whom the unbeatable security system is named, is a comic book artist who has worked on the X-Men series.