The name on the world's best-known toy bricks is older than the bricks themselves. It was chosen for a small workshop making wooden toys.
Ole Kirk Christiansen was a carpenter in Billund, Denmark. He began making wooden toys in 1932. In 1934 he needed a name for the business. According to Wikipedia's history of the company, he held a competition among his staff, with a bottle of homemade wine as the prize. The company's own history says the name was officially in use from January 1936.
The winning name took the first two letters of each word in the Danish phrase leg godt, "play well." Christiansen also considered Legio, suggesting a "legion" of toys, before settling on Lego.
The famous coincidence came later. In Latin, lego is a real verb. Etymonline glosses it as "I study" or "I put together." Wikipedia says the Lego Group later found it loosely means "I assemble." Etymonline says Christiansen did not realize this until later. For a firm that became known for things that snap together, it was a lucky accident rather than a plan.
The bricks arrived well after the name. The company began making plastic toys in 1947. In 1949 it produced an early interlocking brick sold as "Automatic Binding Bricks," based on the Kiddicraft Self-Locking Bricks designed by Hilary Page. The modern stud-and-tube brick followed in 1958, when a patent application was filed in Denmark on January 28.
In English dictionaries, Etymonline dates Lego as a proprietary name to 1954, while noting the company's own claim of use since 1934.
The name still causes small arguments. Many speakers treat it as a mass noun and talk about "some Lego." American English also allows the plural "Legos." The Lego Group asks for neither. It wants the name used as an adjective, as in "LEGO bricks," in line with its trademark.
The company motto dates from 1936, two years after the name. In Danish it is det bedste er ikke for godt, "only the best is good enough." It carries the same godt as leg godt.