The name Lego is normally said to be derived from the Danish words for 'play well': leg godt.
Ole Kirk Christiansen's third son was called Godtfred, and he was Managing Director of The Lego Group from 1957 to 1979. I have heard it said that the plastic bricks that made Lego famous were invented by Ole and Godtfred between them. If this were true, it would give another very neat explanation for why they chose the name (it's a cryptogram formed from their two names). Godtfred was 12 years old when his father began making wooden toys, and 14 when he named the company; he became Junior Managing Director in 1950 (on his 30th birthday). Lego launched its plastic bricks in 1958 (based on a similar product which was patented in the UK in 1939 and launched in 1947, by Kiddicraft), so it is eminently plausible that Godtfred helped to develop the Lego product. It would seem equally plausible that Ole was inspired by his son's name, and the cryptogram suggested by concatenating it with his own, when he named his company. But I can't find any mention of this in any primary source.
Lego also means 'I read' in Latin, but I think this can only be a coincidence. (According to an article in The Guardian, posted in January 2000 – announcing Lego as the British Association of Toy Retailers' choice of Lego as the Toy of the Century – it means 'to put together' in Latin. According to Google Translate however, the Latin for 'put together' – more precisely, I think, 'I put together' – is coniungo.)
© Haydn Thompson 2017