Though it sounds a bit like a fake name, there apparently really was an Archie McPhee. According to the Archie McPhee website (http://www.archiemcphee.com/info/history.html), he was the great-uncle-in-law of the current owner of the company. In 1924, he traveled to China and Japan with a jazz band he formed called the North Star Music Makers. Since he was only the band's manager, passage for him was not booked aboard the S.S. President Madison, but one of the ship's firemen never showed up for the trip, so Archie took his job shoveling coal into the ship's furnaces. He managed the band from the hold of the ship.

After playing gigs at hotels and cafes in Shanghai, Kobe, Yokohama, and Manila, the band was offered a six-month contract playing at the Army and Navy Club in the Philippines. After that, they went home.

The website never explains why this particular escapade made him worthy of having a pop-kitsch toy store named after him.

Addendum: Demeter informs me that Archie McPhee was the name of the campy chef in the kitschy British soap opera "Crossroads". This may be the true origin of the store's name...