Balilla was a brand name used by Giovanni Galli, largely for brakes but also for a single derailleur model. The brand was possibly established sometime in the 1930s in Torino, Italy.

The Balilla brand was more than a little politically charged. Balilla was the nickname of Giovan Battista Perasso, an 18th century figure who was adopted by the Mussolini's Fascists as a youth hero. The Italian version of the Hitler Youth was called the ONB, the Opera Nazionale Balilla.

It is not completely clear to me how the Balilla brand relates to the Galli brand. Balilla was a not a low-end brand, Ferdi Kubler won the 1950 Tour de France using Balilla brakes. It is possible that Giovanni Galli used the Galli brand to effectively replace the Balilla brand sometime in the 1950s, 1960s or even 1970s.

I would note that Giovanni Galli first patented a derailleur design in 1933. This is early, even compared to the offerings of Vittoria and Campagnolo. This Balilla design is a detailed and properly developed rod type mechanism, a bit reminiscent of the early Campagnolo rod-type devices. I have never seen an example of it in the flesh.

However the Balilla derailleur that made it into production was a fairly basic item - a single pulley design not unlike the Simplex Champion du Monde. The design was licensed from Vittoria. Some people claim that Vittoria manufactured it for Balilla.

As you can tell, I could use some more information here!