Falcon Field got its start prior to World War II, when Hollywood producer Leland Hayward and pilot John H. "Jack" Connelly founded Southwest Airways with funding from friends like Henry Fonda, Fred Astaire, Ginger Rogers, James Stewart, Hoagy Carmichael and others. Southwest Airways operated two other airfields in Arizona -- Thunderbird Field No. 1 (now the site of Thunderbird School of Global Management) and Thunderbird Field No. 2 (now the site of Scottsdale Airport) -- to train pilots from China, Russia and 24 other Allied nations. Falcon was to be Thunderbird Field III and would train British pilots.
But the British said they'd like the field to be named after one of their birds, and thus Falcon Field was opened as the No. 4 British Flying Training School (BFTS). There were six BFTS airfields in the U.S., in Florida, Oklahoma, Texas, California and Arizona.
--Read the entire article on Wikipedia: Falcon Field (Arizona)
See also the Falcon Field History (PDF) copied from the City of Mesa, Arizona web site.