As with a lot of IKEA product names, this references what the product is: 'bygg' is 'build', and 'lek' is 'play'.
Interestingly Old English apparently has "lac" which belongs to the former group, while "play" is also of Germanic origin, but derived from words implying moving about and being engaged in rather than directly from the more specific Germanic words for play.
When I looked it up I expected to see one of the more usual examples when there are pairs like this of one Norse/Germanic source and one imported and bastardised from latin or a romance language.